Results of Bessel Functions II

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DLMF Formula Constraints Maple Mathematica Symbolic
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10.22.E38 ∫ 0 1 t ⁢ J ν ⁡ ( α ℓ ⁢ t ) ⁢ J ν ⁡ ( α m ⁢ t ) ⁢ d t = ( a 2 b 2 + α ℓ 2 - ν 2 ) ⁢ ( J ν ⁡ ( α ℓ ) ) 2 2 ⁢ α ℓ 2 ⁢ δ ℓ , m superscript subscript 0 1 𝑡 Bessel-J 𝜈 subscript 𝛼 ℓ 𝑡 Bessel-J 𝜈 subscript 𝛼 𝑚 𝑡 𝑡 superscript 𝑎 2 superscript 𝑏 2 superscript subscript 𝛼 ℓ 2 superscript 𝜈 2 superscript Bessel-J 𝜈 subscript 𝛼 ℓ 2 2 superscript subscript 𝛼 ℓ 2 Kronecker ℓ 𝑚 {\displaystyle{\displaystyle\int_{0}^{1}tJ_{\nu}\left(\alpha_{\ell}t\right)J_{% \nu}\left(\alpha_{m}t\right)\mathrm{d}t=\left(\frac{a^{2}}{b^{2}}+\alpha_{\ell% }^{2}-\nu^{2}\right)\frac{(J_{\nu}\left(\alpha_{\ell}\right))^{2}}{2\alpha_{% \ell}^{2}}\delta_{\ell,m}}}
\int_{0}^{1}t\BesselJ{\nu}@{\alpha_{\ell}t}\BesselJ{\nu}@{\alpha_{m}t}\diff{t} = \left(\frac{a^{2}}{b^{2}}+\alpha_{\ell}^{2}-\nu^{2}\right)\frac{(\BesselJ{\nu}@{\alpha_{\ell}})^{2}}{2\alpha_{\ell}^{2}}\Kroneckerdelta{\ell}{m}		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
int(t*BesselJ(nu, alpha[ell]*t)*BesselJ(nu, alpha[m]*t), t = 0..1) = (((a)^(2))/((b)^(2))+ (alpha[ell])^(2)- (nu)^(2))*((BesselJ(nu, alpha[ell]))^(2))/(2*(alpha[ell])^(2))*KroneckerDelta[ell, m]

Integrate[t*BesselJ[\[Nu], Subscript[\[Alpha], \[ScriptL]]*t]*BesselJ[\[Nu], Subscript[\[Alpha], m]*t], {t, 0, 1}, GenerateConditions->None] == (Divide[(a)^(2),(b)^(2)]+ (Subscript[\[Alpha], \[ScriptL]])^(2)- \[Nu]^(2))*Divide[(BesselJ[\[Nu], Subscript[\[Alpha], \[ScriptL]]])^(2),2*(Subscript[\[Alpha], \[ScriptL]])^(2)]*KroneckerDelta[\[ScriptL], m]
Failure Failure Error
Failed [300 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[m, 1], Rule[α, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[α, m], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[α, ℓ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[m, 2], Rule[α, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[α, m], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[α, ℓ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.22.E39 ∫ x ∞ J 0 ⁡ ( t ) t ⁢ d t + γ + ln ⁡ ( 1 2 ⁢ x ) = ∫ 0 x 1 - J 0 ⁡ ( t ) t ⁢ d t superscript subscript 𝑥 Bessel-J 0 𝑡 𝑡 𝑡 1 2 𝑥 superscript subscript 0 𝑥 1 Bessel-J 0 𝑡 𝑡 𝑡 {\displaystyle{\displaystyle\int_{x}^{\infty}\frac{J_{0}\left(t\right)}{t}% \mathrm{d}t+\gamma+\ln\left(\tfrac{1}{2}x\right)=\int_{0}^{x}\frac{1-J_{0}% \left(t\right)}{t}\mathrm{d}t}}
\int_{x}^{\infty}\frac{\BesselJ{0}@{t}}{t}\diff{t}+\EulerConstant+\ln@{\tfrac{1}{2}x} = \int_{0}^{x}\frac{1-\BesselJ{0}@{t}}{t}\diff{t}		 ℜ ⁡ ( 0 + k + 1 ) > 0 0 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0}} 
int((BesselJ(0, t))/(t), t = x..infinity)+ gamma + ln((1)/(2)*x) = int((1 - BesselJ(0, t))/(t), t = 0..x)

Integrate[Divide[BesselJ[0, t],t], {t, x, Infinity}, GenerateConditions->None]+ EulerGamma + Log[Divide[1,2]*x] == Integrate[Divide[1 - BesselJ[0, t],t], {t, 0, x}, GenerateConditions->None]
Successful Successful - Successful [Tested: 3]
10.22.E39 ∫ 0 x 1 - J 0 ⁡ ( t ) t ⁢ d t = ∑ k = 1 ∞ ( - 1 ) k - 1 ⁢ ( 1 2 ⁢ x ) 2 ⁢ k 2 ⁢ k ⁢ ( k ! ) 2 superscript subscript 0 𝑥 1 Bessel-J 0 𝑡 𝑡 𝑡 superscript subscript 𝑘 1 superscript 1 𝑘 1 superscript 1 2 𝑥 2 𝑘 2 𝑘 superscript 𝑘 2 {\displaystyle{\displaystyle\int_{0}^{x}\frac{1-J_{0}\left(t\right)}{t}\mathrm% {d}t=\sum_{k=1}^{\infty}(-1)^{k-1}\frac{(\frac{1}{2}x)^{2k}}{2k(k!)^{2}}}}
\int_{0}^{x}\frac{1-\BesselJ{0}@{t}}{t}\diff{t} = \sum_{k=1}^{\infty}(-1)^{k-1}\frac{(\frac{1}{2}x)^{2k}}{2k(k!)^{2}}		 ℜ ⁡ ( 0 + k + 1 ) > 0 0 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0}} 
int((1 - BesselJ(0, t))/(t), t = 0..x) = sum((- 1)^(k - 1)*(((1)/(2)*x)^(2*k))/(2*k*(factorial(k))^(2)), k = 1..infinity)

Integrate[Divide[1 - BesselJ[0, t],t], {t, 0, x}, GenerateConditions->None] == Sum[(- 1)^(k - 1)*Divide[(Divide[1,2]*x)^(2*k),2*k*((k)!)^(2)], {k, 1, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 3]
10.22.E40 ∫ x ∞ Y 0 ⁡ ( t ) t ⁢ d t = - 1 π ⁢ ( ln ⁡ ( 1 2 ⁢ x ) + γ ) 2 + π 6 + 2 π ⁢ ∑ k = 1 ∞ ( - 1 ) k ⁢ ( ψ ⁡ ( k + 1 ) + 1 2 ⁢ k - ln ⁡ ( 1 2 ⁢ x ) ) ⁢ ( 1 2 ⁢ x ) 2 ⁢ k 2 ⁢ k ⁢ ( k ! ) 2 superscript subscript 𝑥 Bessel-Y-Weber 0 𝑡 𝑡 𝑡 1 𝜋 superscript 1 2 𝑥 2 𝜋 6 2 𝜋 superscript subscript 𝑘 1 superscript 1 𝑘 digamma 𝑘 1 1 2 𝑘 1 2 𝑥 superscript 1 2 𝑥 2 𝑘 2 𝑘 superscript 𝑘 2 {\displaystyle{\displaystyle\int_{x}^{\infty}\frac{Y_{0}\left(t\right)}{t}% \mathrm{d}t=-\frac{1}{\pi}\left(\ln\left(\tfrac{1}{2}x\right)+\gamma\right)^{2% }+\frac{\pi}{6}+\frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\*\left(\psi\left(k+1% \right)+\frac{1}{2k}-\ln\left(\tfrac{1}{2}x\right)\right)\frac{(\tfrac{1}{2}x)% ^{2k}}{2k(k!)^{2}}}}
\int_{x}^{\infty}\frac{\BesselY{0}@{t}}{t}\diff{t} = -\frac{1}{\pi}\left(\ln@{\tfrac{1}{2}x}+\EulerConstant\right)^{2}+\frac{\pi}{6}+\frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\*\left(\digamma@{k+1}+\frac{1}{2k}-\ln@{\tfrac{1}{2}x}\right)\frac{(\tfrac{1}{2}x)^{2k}}{2k(k!)^{2}}		 ℜ ⁡ ( 0 + k + 1 ) > 0 , ℜ ⁡ ( ( - 0 ) + k + 1 ) > 0 formulae-sequence 0 𝑘 1 0 0 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0,\Re((-0)+k+1)>0}} 
int((BesselY(0, t))/(t), t = x..infinity) = -(1)/(Pi)*(ln((1)/(2)*x)+ gamma)^(2)+(Pi)/(6)+(2)/(Pi)*sum((- 1)^(k)*(Psi(k + 1)+(1)/(2*k)- ln((1)/(2)*x))*(((1)/(2)*x)^(2*k))/(2*k*(factorial(k))^(2)), k = 1..infinity)

Integrate[Divide[BesselY[0, t],t], {t, x, Infinity}, GenerateConditions->None] == -Divide[1,Pi]*(Log[Divide[1,2]*x]+ EulerGamma)^(2)+Divide[Pi,6]+Divide[2,Pi]*Sum[(- 1)^(k)*(PolyGamma[k + 1]+Divide[1,2*k]- Log[Divide[1,2]*x])*Divide[(Divide[1,2]*x)^(2*k),2*k*((k)!)^(2)], {k, 1, Infinity}, GenerateConditions->None]
Aborted Aborted Skipped - Because timed out Skipped - Because timed out
10.22.E41 ∫ 0 ∞ J ν ⁡ ( t ) ⁢ d t = 1 superscript subscript 0 Bessel-J 𝜈 𝑡 𝑡 1 {\displaystyle{\displaystyle\int_{0}^{\infty}J_{\nu}\left(t\right)\mathrm{d}t=% 1}}
\int_{0}^{\infty}\BesselJ{\nu}@{t}\diff{t} = 1		 ℜ ⁡ ν > - 1 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 1 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\nu>-1,\Re(\nu+k+1)>0}} 
int(BesselJ(nu, t), t = 0..infinity) = 1

Integrate[BesselJ[\[Nu], t], {t, 0, Infinity}, GenerateConditions->None] == 1
Successful Successful - Successful [Tested: 8]
10.22.E42 ∫ 0 ∞ Y ν ⁡ ( t ) ⁢ d t = - tan ⁡ ( 1 2 ⁢ ν ⁢ π ) superscript subscript 0 Bessel-Y-Weber 𝜈 𝑡 𝑡 1 2 𝜈 𝜋 {\displaystyle{\displaystyle\int_{0}^{\infty}Y_{\nu}\left(t\right)\mathrm{d}t=% -\tan\left(\tfrac{1}{2}\nu\pi\right)}}
\int_{0}^{\infty}\BesselY{\nu}@{t}\diff{t} = -\tan@{\tfrac{1}{2}\nu\pi}		 | ℜ ⁡ ν | < 1 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 formulae-sequence 𝜈 1 formulae-sequence 𝜈 𝑘 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle|\Re\nu|<1,\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}} 
int(BesselY(nu, t), t = 0..infinity) = - tan((1)/(2)*nu*Pi)

Integrate[BesselY[\[Nu], t], {t, 0, Infinity}, GenerateConditions->None] == - Tan[Divide[1,2]*\[Nu]*Pi]
Successful Aborted - Successful [Tested: 6]
10.22.E43 ∫ 0 ∞ t μ ⁢ J ν ⁡ ( t ) ⁢ d t = 2 μ ⁢ Γ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ + 1 2 ) Γ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ μ + 1 2 ) superscript subscript 0 superscript 𝑡 𝜇 Bessel-J 𝜈 𝑡 𝑡 superscript 2 𝜇 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 2 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 2 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu}J_{\nu}\left(t\right)% \mathrm{d}t=2^{\mu}\frac{\Gamma\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{1}% {2}\right)}{\Gamma\left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+\tfrac{1}{2}\right)}}}
\int_{0}^{\infty}t^{\mu}\BesselJ{\nu}@{t}\diff{t} = 2^{\mu}\frac{\EulerGamma@{\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{1}{2}}}{\EulerGamma@{\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+\tfrac{1}{2}}}		 ℜ ⁡ ( μ + ν ) > - 1 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ + 1 2 ) > 0 , ℜ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ μ + 1 2 ) > 0 formulae-sequence 𝜇 𝜈 1 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 1 2 𝜈 1 2 𝜇 1 2 0 1 2 𝜈 1 2 𝜇 1 2 0 {\displaystyle{\displaystyle\Re\left(\mu+\nu\right)>-1,\Re(\nu+k+1)>0,\Re(% \tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{1}{2})>0,\Re(\tfrac{1}{2}\nu-\tfrac{1}{% 2}\mu+\tfrac{1}{2})>0}} 
int((t)^(mu)* BesselJ(nu, t), t = 0..infinity) = (2)^(mu)*(GAMMA((1)/(2)*nu +(1)/(2)*mu +(1)/(2)))/(GAMMA((1)/(2)*nu -(1)/(2)*mu +(1)/(2)))

Integrate[(t)^\[Mu]* BesselJ[\[Nu], t], {t, 0, Infinity}, GenerateConditions->None] == (2)^\[Mu]*Divide[Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2]],Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2]]]
Successful Successful - Successful [Tested: 10]
10.22.E44 ∫ 0 ∞ t μ ⁢ Y ν ⁡ ( t ) ⁢ d t = 2 μ π ⁢ Γ ⁡ ( 1 2 ⁢ μ + 1 2 ⁢ ν + 1 2 ) ⁢ Γ ⁡ ( 1 2 ⁢ μ - 1 2 ⁢ ν + 1 2 ) ⁢ sin ⁡ ( 1 2 ⁢ μ - 1 2 ⁢ ν ) ⁢ π superscript subscript 0 superscript 𝑡 𝜇 Bessel-Y-Weber 𝜈 𝑡 𝑡 superscript 2 𝜇 𝜋 Euler-Gamma 1 2 𝜇 1 2 𝜈 1 2 Euler-Gamma 1 2 𝜇 1 2 𝜈 1 2 1 2 𝜇 1 2 𝜈 𝜋 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu}Y_{\nu}\left(t\right)% \mathrm{d}t=\frac{2^{\mu}}{\pi}\Gamma\left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+% \tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}% \right)\sin\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu\right)\pi}}
\int_{0}^{\infty}t^{\mu}\BesselY{\nu}@{t}\diff{t} = \frac{2^{\mu}}{\pi}\EulerGamma@{\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2}}\EulerGamma@{\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}}\sin@{\tfrac{1}{2}\mu-\tfrac{1}{2}\nu}\pi		 ℜ ⁡ ( μ + ν ) > - 1 , ℜ ⁡ ( μ - ν ) > - 1 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 , ℜ ⁡ ( 1 2 ⁢ μ + 1 2 ⁢ ν + 1 2 ) > 0 , ℜ ⁡ ( 1 2 ⁢ μ - 1 2 ⁢ ν + 1 2 ) > 0 formulae-sequence 𝜇 𝜈 1 formulae-sequence 𝜇 𝜈 1 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 1 2 𝜇 1 2 𝜈 1 2 0 1 2 𝜇 1 2 𝜈 1 2 0 {\displaystyle{\displaystyle\Re\left(\mu+\nu\right)>-1,\Re\left(\mu-\nu\right)% >-1,\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+% \tfrac{1}{2})>0,\Re(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2})>0}} 
int((t)^(mu)* BesselY(nu, t), t = 0..infinity) = ((2)^(mu))/(Pi)*GAMMA((1)/(2)*mu +(1)/(2)*nu +(1)/(2))*GAMMA((1)/(2)*mu -(1)/(2)*nu +(1)/(2))*sin((1)/(2)*mu -(1)/(2)*nu)*Pi

Integrate[(t)^\[Mu]* BesselY[\[Nu], t], {t, 0, Infinity}, GenerateConditions->None] == Divide[(2)^\[Mu],Pi]*Gamma[Divide[1,2]*\[Mu]+Divide[1,2]*\[Nu]+Divide[1,2]]*Gamma[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]+Divide[1,2]]*Sin[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]]*Pi
Error Aborted -
Failed [10 / 10]
Result: Complex[-0.5512405929316078, 0.2551977660147906]
Test Values: {Rule[μ, 0], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.26217720344291356, -0.18052742798771904]
Test Values: {Rule[μ, 0], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.22.E45 ∫ 0 ∞ 1 - J 0 ⁡ ( t ) t μ ⁢ d t = - π ⁢ sec ⁡ ( 1 2 ⁢ μ ⁢ π ) 2 μ ⁢ Γ 2 ⁡ ( 1 2 ⁢ μ + 1 2 ) superscript subscript 0 1 Bessel-J 0 𝑡 superscript 𝑡 𝜇 𝑡 𝜋 1 2 𝜇 𝜋 superscript 2 𝜇 Euler-Gamma 2 1 2 𝜇 1 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{1-J_{0}\left(t\right)}{t^{% \mu}}\mathrm{d}t=-\frac{\pi\sec\left(\frac{1}{2}\mu\pi\right)}{2^{\mu}{\Gamma^% {2}}\left(\frac{1}{2}\mu+\frac{1}{2}\right)}}}
\int_{0}^{\infty}\frac{1-\BesselJ{0}@{t}}{t^{\mu}}\diff{t} = -\frac{\pi\sec@{\frac{1}{2}\mu\pi}}{2^{\mu}\EulerGamma^{2}@{\frac{1}{2}\mu+\frac{1}{2}}}		 1 < ℜ ⁡ μ , ℜ ⁡ μ < 3 , ℜ ⁡ ( 0 + k + 1 ) > 0 , ℜ ⁡ ( 1 2 ⁢ μ + 1 2 ) > 0 formulae-sequence 1 𝜇 formulae-sequence 𝜇 3 formulae-sequence 0 𝑘 1 0 1 2 𝜇 1 2 0 {\displaystyle{\displaystyle 1<\Re\mu,\Re\mu<3,\Re(0+k+1)>0,\Re(\frac{1}{2}\mu% +\frac{1}{2})>0}} 
int((1 - BesselJ(0, t))/((t)^(mu)), t = 0..infinity) = -(Pi*sec((1)/(2)*mu*Pi))/((2)^(mu)* (GAMMA((1)/(2)*mu +(1)/(2)))^(2))

Integrate[Divide[1 - BesselJ[0, t],(t)^\[Mu]], {t, 0, Infinity}, GenerateConditions->None] == -Divide[Pi*Sec[Divide[1,2]*\[Mu]*Pi],(2)^\[Mu]* (Gamma[Divide[1,2]*\[Mu]+Divide[1,2]])^(2)]
Error Aborted - Successful [Tested: 10]
10.22.E46 ∫ 0 ∞ t ν + 1 ⁢ J ν ⁡ ( a ⁢ t ) ( t 2 + b 2 ) μ + 1 ⁢ d t = a μ ⁢ b ν - μ 2 μ ⁢ Γ ⁡ ( μ + 1 ) ⁢ K ν - μ ⁡ ( a ⁢ b ) superscript subscript 0 superscript 𝑡 𝜈 1 Bessel-J 𝜈 𝑎 𝑡 superscript superscript 𝑡 2 superscript 𝑏 2 𝜇 1 𝑡 superscript 𝑎 𝜇 superscript 𝑏 𝜈 𝜇 superscript 2 𝜇 Euler-Gamma 𝜇 1 modified-Bessel-second-kind 𝜈 𝜇 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{t^{\nu+1}J_{\nu}\left(at% \right)}{(t^{2}+b^{2})^{\mu+1}}\mathrm{d}t=\frac{a^{\mu}b^{\nu-\mu}}{2^{\mu}% \Gamma\left(\mu+1\right)}K_{\nu-\mu}\left(ab\right)}}
\int_{0}^{\infty}\frac{t^{\nu+1}\BesselJ{\nu}@{at}}{(t^{2}+b^{2})^{\mu+1}}\diff{t} = \frac{a^{\mu}b^{\nu-\mu}}{2^{\mu}\EulerGamma@{\mu+1}}\modBesselK{\nu-\mu}@{ab}		 a > 0 , ℜ ⁡ b > 0 , - 1 < ℜ ⁡ ν , ℜ ⁡ ν < 2 ⁢ ℜ ⁡ μ + 3 2 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( μ + 1 ) > 0 formulae-sequence 𝑎 0 formulae-sequence 𝑏 0 formulae-sequence 1 𝜈 formulae-sequence 𝜈 2 𝜇 3 2 formulae-sequence 𝜈 𝑘 1 0 𝜇 1 0 {\displaystyle{\displaystyle a>0,\Re b>0,-1<\Re\nu,\Re\nu<2\Re\mu+\tfrac{3}{2}% ,\Re(\nu+k+1)>0,\Re(\mu+1)>0}} 
int(((t)^(nu + 1)* BesselJ(nu, a*t))/(((t)^(2)+ (b)^(2))^(mu + 1)), t = 0..infinity) = ((a)^(mu)* (b)^(nu - mu))/((2)^(mu)* GAMMA(mu + 1))*BesselK(nu - mu, a*b)

Integrate[Divide[(t)^(\[Nu]+ 1)* BesselJ[\[Nu], a*t],((t)^(2)+ (b)^(2))^(\[Mu]+ 1)], {t, 0, Infinity}, GenerateConditions->None] == Divide[(a)^\[Mu]* (b)^(\[Nu]- \[Mu]),(2)^\[Mu]* Gamma[\[Mu]+ 1]]*BesselK[\[Nu]- \[Mu], a*b]
Error Aborted - Skipped - Because timed out
10.22.E47 ∫ 0 ∞ t ν ⁢ Y ν ⁡ ( a ⁢ t ) t 2 + b 2 ⁢ d t = - b ν - 1 ⁢ K ν ⁡ ( a ⁢ b ) superscript subscript 0 superscript 𝑡 𝜈 Bessel-Y-Weber 𝜈 𝑎 𝑡 superscript 𝑡 2 superscript 𝑏 2 𝑡 superscript 𝑏 𝜈 1 modified-Bessel-second-kind 𝜈 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{t^{\nu}Y_{\nu}\left(at% \right)}{t^{2}+b^{2}}\mathrm{d}t=-b^{\nu-1}K_{\nu}\left(ab\right)}}
\int_{0}^{\infty}\frac{t^{\nu}\BesselY{\nu}@{at}}{t^{2}+b^{2}}\diff{t} = -b^{\nu-1}\modBesselK{\nu}@{ab}		 a > 0 , ℜ ⁡ b > 0 , - 1 2 < ℜ ⁡ ν , ℜ ⁡ ν < 5 2 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 formulae-sequence 𝑎 0 formulae-sequence 𝑏 0 formulae-sequence 1 2 𝜈 formulae-sequence 𝜈 5 2 formulae-sequence 𝜈 𝑘 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle a>0,\Re b>0,-\tfrac{1}{2}<\Re\nu,\Re\nu<\tfrac{5}% {2},\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}} 
int(((t)^(nu)* BesselY(nu, a*t))/((t)^(2)+ (b)^(2)), t = 0..infinity) = - (b)^(nu - 1)* BesselK(nu, a*b)

Integrate[Divide[(t)^\[Nu]* BesselY[\[Nu], a*t],(t)^(2)+ (b)^(2)], {t, 0, Infinity}, GenerateConditions->None] == - (b)^(\[Nu]- 1)* BesselK[\[Nu], a*b]
Error Aborted - Skipped - Because timed out
10.22.E48 ∫ 0 ∞ J μ ⁡ ( x ⁢ cosh ⁡ ϕ ) ⁢ ( cosh ⁡ ϕ ) 1 - μ ⁢ ( sinh ⁡ ϕ ) 2 ⁢ ν + 1 ⁢ d ϕ = 2 ν ⁢ Γ ⁡ ( ν + 1 ) ⁢ x - ν - 1 ⁢ J μ - ν - 1 ⁡ ( x ) superscript subscript 0 Bessel-J 𝜇 𝑥 italic-ϕ superscript italic-ϕ 1 𝜇 superscript italic-ϕ 2 𝜈 1 italic-ϕ superscript 2 𝜈 Euler-Gamma 𝜈 1 superscript 𝑥 𝜈 1 Bessel-J 𝜇 𝜈 1 𝑥 {\displaystyle{\displaystyle\int_{0}^{\infty}J_{\mu}\left(x\cosh\phi\right)(% \cosh\phi)^{1-\mu}(\sinh\phi)^{2\nu+1}\mathrm{d}\phi=2^{\nu}\Gamma\left(\nu+1% \right)x^{-\nu-1}J_{\mu-\nu-1}\left(x\right)}}
\int_{0}^{\infty}\BesselJ{\mu}@{x\cosh@@{\phi}}(\cosh@@{\phi})^{1-\mu}(\sinh@@{\phi})^{2\nu+1}\diff{\phi} = 2^{\nu}\EulerGamma@{\nu+1}x^{-\nu-1}\BesselJ{\mu-\nu-1}@{x}		 x > 0 , ℜ ⁡ ν > - 1 , ℜ ⁡ μ > 2 ⁢ ℜ ⁡ ν + 1 2 , ℜ ⁡ ( ( μ ) + k + 1 ) > 0 , ℜ ⁡ ( ( μ - ν - 1 ) + k + 1 ) > 0 , ℜ ⁡ ( ν + 1 ) > 0 formulae-sequence 𝑥 0 formulae-sequence 𝜈 1 formulae-sequence 𝜇 2 𝜈 1 2 formulae-sequence 𝜇 𝑘 1 0 formulae-sequence 𝜇 𝜈 1 𝑘 1 0 𝜈 1 0 {\displaystyle{\displaystyle x>0,\Re\nu>-1,\Re\mu>2\Re\nu+\tfrac{1}{2},\Re((% \mu)+k+1)>0,\Re((\mu-\nu-1)+k+1)>0,\Re(\nu+1)>0}} 
int(BesselJ(mu, x*cosh(phi))*(cosh(phi))^(1 - mu)*(sinh(phi))^(2*nu + 1), phi = 0..infinity) = (2)^(nu)* GAMMA(nu + 1)*(x)^(- nu - 1)* BesselJ(mu - nu - 1, x)

Integrate[BesselJ[\[Mu], x*Cosh[\[Phi]]]*(Cosh[\[Phi]])^(1 - \[Mu])*(Sinh[\[Phi]])^(2*\[Nu]+ 1), {\[Phi], 0, Infinity}, GenerateConditions->None] == (2)^\[Nu]* Gamma[\[Nu]+ 1]*(x)^(- \[Nu]- 1)* BesselJ[\[Mu]- \[Nu]- 1, x]
Error Aborted - Skipped - Because timed out
10.22.E49 ∫ 0 ∞ t μ - 1 ⁢ e - a ⁢ t ⁢ J ν ⁡ ( b ⁢ t ) ⁢ d t = ( 1 2 ⁢ b ) ν a μ + ν ⁢ Γ ⁡ ( μ + ν ) ⁢ 𝐅 ⁡ ( μ + ν 2 , μ + ν + 1 2 ; ν + 1 ; - b 2 a 2 ) superscript subscript 0 superscript 𝑡 𝜇 1 superscript 𝑒 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 𝑡 superscript 1 2 𝑏 𝜈 superscript 𝑎 𝜇 𝜈 Euler-Gamma 𝜇 𝜈 scaled-hypergeometric-bold-F 𝜇 𝜈 2 𝜇 𝜈 1 2 𝜈 1 superscript 𝑏 2 superscript 𝑎 2 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu-1}e^{-at}J_{\nu}\left(bt% \right)\mathrm{d}t=\frac{(\tfrac{1}{2}b)^{\nu}}{a^{\mu+\nu}}\Gamma\left(\mu+% \nu\right)\*\mathbf{F}\left(\frac{\mu+\nu}{2},\frac{\mu+\nu+1}{2};\nu+1;-\frac% {b^{2}}{a^{2}}\right)}}
\int_{0}^{\infty}t^{\mu-1}e^{-at}\BesselJ{\nu}@{bt}\diff{t} = \frac{(\tfrac{1}{2}b)^{\nu}}{a^{\mu+\nu}}\EulerGamma@{\mu+\nu}\*\hyperOlverF@{\frac{\mu+\nu}{2}}{\frac{\mu+\nu+1}{2}}{\nu+1}{-\frac{b^{2}}{a^{2}}}		 ℜ ⁡ ( μ + ν ) > 0 , ℜ ⁡ ( a + i ⁢ b ) > 0 , ℜ ⁡ ( a - i ⁢ b ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( μ + ν ) > 0 formulae-sequence 𝜇 𝜈 0 formulae-sequence 𝑎 𝑖 𝑏 0 formulae-sequence 𝑎 𝑖 𝑏 0 formulae-sequence 𝜈 𝑘 1 0 𝜇 𝜈 0 {\displaystyle{\displaystyle\Re\left(\mu+\nu\right)>0,\Re\left(a+ib\right)>0,% \Re\left(a-ib\right)>0,\Re(\nu+k+1)>0,\Re(\mu+\nu)>0}} 
int((t)^(mu - 1)* exp(- a*t)*BesselJ(nu, b*t), t = 0..infinity) = (((1)/(2)*b)^(nu))/((a)^(mu + nu))*GAMMA(mu + nu)* hypergeom([(mu + nu)/(2), (mu + nu + 1)/(2)], [nu + 1], -((b)^(2))/((a)^(2)))/GAMMA(nu + 1)

Integrate[(t)^(\[Mu]- 1)* Exp[- a*t]*BesselJ[\[Nu], b*t], {t, 0, Infinity}, GenerateConditions->None] == Divide[(Divide[1,2]*b)^\[Nu],(a)^(\[Mu]+ \[Nu])]*Gamma[\[Mu]+ \[Nu]]* Hypergeometric2F1Regularized[Divide[\[Mu]+ \[Nu],2], Divide[\[Mu]+ \[Nu]+ 1,2], \[Nu]+ 1, -Divide[(b)^(2),(a)^(2)]]
Error Aborted - Successful [Tested: 0]
10.22.E50 ∫ 0 ∞ t μ - 1 ⁢ e - a ⁢ t ⁢ Y ν ⁡ ( b ⁢ t ) ⁢ d t = cot ⁡ ( ν ⁢ π ) ⁢ ( 1 2 ⁢ b ) ν ⁢ Γ ⁡ ( μ + ν ) ( a 2 + b 2 ) 1 2 ⁢ ( μ + ν ) ⁢ 𝐅 ⁡ ( μ + ν 2 , 1 - μ + ν 2 ; ν + 1 ; b 2 a 2 + b 2 ) - csc ⁡ ( ν ⁢ π ) ⁢ ( 1 2 ⁢ b ) - ν ⁢ Γ ⁡ ( μ - ν ) ( a 2 + b 2 ) 1 2 ⁢ ( μ - ν ) ⁢ 𝐅 ⁡ ( μ - ν 2 , 1 - μ - ν 2 ; 1 - ν ; b 2 a 2 + b 2 ) superscript subscript 0 superscript 𝑡 𝜇 1 superscript 𝑒 𝑎 𝑡 Bessel-Y-Weber 𝜈 𝑏 𝑡 𝑡 𝜈 𝜋 superscript 1 2 𝑏 𝜈 Euler-Gamma 𝜇 𝜈 superscript superscript 𝑎 2 superscript 𝑏 2 1 2 𝜇 𝜈 scaled-hypergeometric-bold-F 𝜇 𝜈 2 1 𝜇 𝜈 2 𝜈 1 superscript 𝑏 2 superscript 𝑎 2 superscript 𝑏 2 𝜈 𝜋 superscript 1 2 𝑏 𝜈 Euler-Gamma 𝜇 𝜈 superscript superscript 𝑎 2 superscript 𝑏 2 1 2 𝜇 𝜈 scaled-hypergeometric-bold-F 𝜇 𝜈 2 1 𝜇 𝜈 2 1 𝜈 superscript 𝑏 2 superscript 𝑎 2 superscript 𝑏 2 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu-1}e^{-at}Y_{\nu}\left(bt% \right)\mathrm{d}t=\cot\left(\nu\pi\right)\frac{(\tfrac{1}{2}b)^{\nu}\Gamma% \left(\mu+\nu\right)}{(a^{2}+b^{2})^{\frac{1}{2}(\mu+\nu)}}\*\mathbf{F}\left(% \frac{\mu+\nu}{2},\frac{1-\mu+\nu}{2};\nu+1;\frac{b^{2}}{a^{2}+b^{2}}\right)-% \csc\left(\nu\pi\right)\frac{(\tfrac{1}{2}b)^{-\nu}\Gamma\left(\mu-\nu\right)}% {(a^{2}+b^{2})^{\frac{1}{2}(\mu-\nu)}}\*\mathbf{F}\left(\frac{\mu-\nu}{2},% \frac{1-\mu-\nu}{2};1-\nu;\frac{b^{2}}{a^{2}+b^{2}}\right)}}
\int_{0}^{\infty}t^{\mu-1}e^{-at}\BesselY{\nu}@{bt}\diff{t} = \cot@{\nu\pi}\frac{(\tfrac{1}{2}b)^{\nu}\EulerGamma@{\mu+\nu}}{(a^{2}+b^{2})^{\frac{1}{2}(\mu+\nu)}}\*\hyperOlverF@{\frac{\mu+\nu}{2}}{\frac{1-\mu+\nu}{2}}{\nu+1}{\frac{b^{2}}{a^{2}+b^{2}}}-\csc@{\nu\pi}\frac{(\tfrac{1}{2}b)^{-\nu}\EulerGamma@{\mu-\nu}}{(a^{2}+b^{2})^{\frac{1}{2}(\mu-\nu)}}\*\hyperOlverF@{\frac{\mu-\nu}{2}}{\frac{1-\mu-\nu}{2}}{1-\nu}{\frac{b^{2}}{a^{2}+b^{2}}}		 ℜ ⁡ μ > | ℜ ⁡ ν | , ℜ ⁡ ( a + i ⁢ b ) > 0 , ℜ ⁡ ( a - i ⁢ b ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 , ℜ ⁡ ( μ + ν ) > 0 , ℜ ⁡ ( μ - ν ) > 0 formulae-sequence 𝜇 𝜈 formulae-sequence 𝑎 𝑖 𝑏 0 formulae-sequence 𝑎 𝑖 𝑏 0 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 𝜇 𝜈 0 𝜇 𝜈 0 {\displaystyle{\displaystyle\Re\mu>|\Re\nu|,\Re\left(a+ib\right)>0,\Re\left(a-% ib\right)>0,\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re(\mu+\nu)>0,\Re(\mu-\nu)>0}} 
int((t)^(mu - 1)* exp(- a*t)*BesselY(nu, b*t), t = 0..infinity) = cot(nu*Pi)*(((1)/(2)*b)^(nu)* GAMMA(mu + nu))/(((a)^(2)+ (b)^(2))^((1)/(2)*(mu + nu)))* hypergeom([(mu + nu)/(2), (1 - mu + nu)/(2)], [nu + 1], ((b)^(2))/((a)^(2)+ (b)^(2)))/GAMMA(nu + 1)- csc(nu*Pi)*(((1)/(2)*b)^(- nu)* GAMMA(mu - nu))/(((a)^(2)+ (b)^(2))^((1)/(2)*(mu - nu)))* hypergeom([(mu - nu)/(2), (1 - mu - nu)/(2)], [1 - nu], ((b)^(2))/((a)^(2)+ (b)^(2)))/GAMMA(1 - nu)

Integrate[(t)^(\[Mu]- 1)* Exp[- a*t]*BesselY[\[Nu], b*t], {t, 0, Infinity}, GenerateConditions->None] == Cot[\[Nu]*Pi]*Divide[(Divide[1,2]*b)^\[Nu]* Gamma[\[Mu]+ \[Nu]],((a)^(2)+ (b)^(2))^(Divide[1,2]*(\[Mu]+ \[Nu]))]* Hypergeometric2F1Regularized[Divide[\[Mu]+ \[Nu],2], Divide[1 - \[Mu]+ \[Nu],2], \[Nu]+ 1, Divide[(b)^(2),(a)^(2)+ (b)^(2)]]- Csc[\[Nu]*Pi]*Divide[(Divide[1,2]*b)^(- \[Nu])* Gamma[\[Mu]- \[Nu]],((a)^(2)+ (b)^(2))^(Divide[1,2]*(\[Mu]- \[Nu]))]* Hypergeometric2F1Regularized[Divide[\[Mu]- \[Nu],2], Divide[1 - \[Mu]- \[Nu],2], 1 - \[Nu], Divide[(b)^(2),(a)^(2)+ (b)^(2)]]
Error Aborted - Skipped - Because timed out
10.22.E51 ∫ 0 ∞ J ν ⁡ ( b ⁢ t ) ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ t ν + 1 ⁢ d t = b ν ( 2 ⁢ p 2 ) ν + 1 ⁢ exp ⁡ ( - b 2 4 ⁢ p 2 ) superscript subscript 0 Bessel-J 𝜈 𝑏 𝑡 superscript 𝑝 2 superscript 𝑡 2 superscript 𝑡 𝜈 1 𝑡 superscript 𝑏 𝜈 superscript 2 superscript 𝑝 2 𝜈 1 superscript 𝑏 2 4 superscript 𝑝 2 {\displaystyle{\displaystyle\int_{0}^{\infty}J_{\nu}\left(bt\right)\exp\left(-% p^{2}t^{2}\right)t^{\nu+1}\mathrm{d}t=\frac{b^{\nu}}{(2p^{2})^{\nu+1}}\exp% \left(-\frac{b^{2}}{4p^{2}}\right)}}
\int_{0}^{\infty}\BesselJ{\nu}@{bt}\exp@{-p^{2}t^{2}}t^{\nu+1}\diff{t} = \frac{b^{\nu}}{(2p^{2})^{\nu+1}}\exp@{-\frac{b^{2}}{4p^{2}}}		 ℜ ⁡ ν > - 1 , ℜ ⁡ ( p 2 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 1 formulae-sequence superscript 𝑝 2 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\nu>-1,\Re\left(p^{2}\right)>0,\Re(\nu+k+1)>0}} 
int(BesselJ(nu, b*t)*exp(- (p)^(2)* (t)^(2))*(t)^(nu + 1), t = 0..infinity) = ((b)^(nu))/((2*(p)^(2))^(nu + 1))*exp(-((b)^(2))/(4*(p)^(2)))

Integrate[BesselJ[\[Nu], b*t]*Exp[- (p)^(2)* (t)^(2)]*(t)^(\[Nu]+ 1), {t, 0, Infinity}, GenerateConditions->None] == Divide[(b)^\[Nu],(2*(p)^(2))^(\[Nu]+ 1)]*Exp[-Divide[(b)^(2),4*(p)^(2)]]
Error Aborted -
Failed [151 / 300]
Result: Complex[-0.06577510728447342, -0.5886826409090221]
Test Values: {Rule[b, -1.5], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.0556301041786353, -0.2359104145157832]
Test Values: {Rule[b, -1.5], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.22.E52 ∫ 0 ∞ J ν ⁡ ( b ⁢ t ) ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ d t = π 2 ⁢ p ⁢ exp ⁡ ( - b 2 8 ⁢ p 2 ) ⁢ I ν / 2 ⁡ ( b 2 8 ⁢ p 2 ) superscript subscript 0 Bessel-J 𝜈 𝑏 𝑡 superscript 𝑝 2 superscript 𝑡 2 𝑡 𝜋 2 𝑝 superscript 𝑏 2 8 superscript 𝑝 2 modified-Bessel-first-kind 𝜈 2 superscript 𝑏 2 8 superscript 𝑝 2 {\displaystyle{\displaystyle\int_{0}^{\infty}J_{\nu}\left(bt\right)\exp\left(-% p^{2}t^{2}\right)\mathrm{d}t=\frac{\sqrt{\pi}}{2p}\exp\left(-\frac{b^{2}}{8p^{% 2}}\right)I_{\ifrac{\nu}{2}}\left(\frac{b^{2}}{8p^{2}}\right)}}
\int_{0}^{\infty}\BesselJ{\nu}@{bt}\exp@{-p^{2}t^{2}}\diff{t} = \frac{\sqrt{\pi}}{2p}\exp@{-\frac{b^{2}}{8p^{2}}}\modBesselI{\ifrac{\nu}{2}}@{\frac{b^{2}}{8p^{2}}}		 ℜ ⁡ ν > - 1 , ℜ ⁡ ( p 2 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 1 formulae-sequence superscript 𝑝 2 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\nu>-1,\Re\left(p^{2}\right)>0,\Re(\nu+k+1)>0}} 
int(BesselJ(nu, b*t)*exp(- (p)^(2)* (t)^(2)), t = 0..infinity) = (sqrt(Pi))/(2*p)*exp(-((b)^(2))/(8*(p)^(2)))*BesselI((nu)/(2), ((b)^(2))/(8*(p)^(2)))

Integrate[BesselJ[\[Nu], b*t]*Exp[- (p)^(2)* (t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Divide[Sqrt[Pi],2*p]*Exp[-Divide[(b)^(2),8*(p)^(2)]]*BesselI[Divide[\[Nu],2], Divide[(b)^(2),8*(p)^(2)]]
Error Aborted - Skip - No test values generated
10.22.E53 ∫ 0 ∞ Y 2 ⁢ ν ⁡ ( b ⁢ t ) ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ d t = - π 2 ⁢ p ⁢ exp ⁡ ( - b 2 8 ⁢ p 2 ) ⁢ ( I ν ⁡ ( b 2 8 ⁢ p 2 ) ⁢ tan ⁡ ( ν ⁢ π ) + 1 π ⁢ K ν ⁡ ( b 2 8 ⁢ p 2 ) ⁢ sec ⁡ ( ν ⁢ π ) ) superscript subscript 0 Bessel-Y-Weber 2 𝜈 𝑏 𝑡 superscript 𝑝 2 superscript 𝑡 2 𝑡 𝜋 2 𝑝 superscript 𝑏 2 8 superscript 𝑝 2 modified-Bessel-first-kind 𝜈 superscript 𝑏 2 8 superscript 𝑝 2 𝜈 𝜋 1 𝜋 modified-Bessel-second-kind 𝜈 superscript 𝑏 2 8 superscript 𝑝 2 𝜈 𝜋 {\displaystyle{\displaystyle\int_{0}^{\infty}Y_{2\nu}\left(bt\right)\exp\left(% -p^{2}t^{2}\right)\mathrm{d}t=-\frac{\sqrt{\pi}}{2p}\exp\left(-\frac{b^{2}}{8p% ^{2}}\right)\left(I_{\nu}\left(\frac{b^{2}}{8p^{2}}\right)\tan\left(\nu\pi% \right)+\frac{1}{\pi}K_{\nu}\left(\frac{b^{2}}{8p^{2}}\right)\sec\left(\nu\pi% \right)\right)}}
\int_{0}^{\infty}\BesselY{2\nu}@{bt}\exp@{-p^{2}t^{2}}\diff{t} = -\frac{\sqrt{\pi}}{2p}\exp@{-\frac{b^{2}}{8p^{2}}}\left(\modBesselI{\nu}@{\frac{b^{2}}{8p^{2}}}\tan@{\nu\pi}+\frac{1}{\pi}\modBesselK{\nu}@{\frac{b^{2}}{8p^{2}}}\sec@{\nu\pi}\right)		 | ℜ ⁡ ν | < 1 2 , ℜ ⁡ ( p 2 ) > 0 , ℜ ⁡ ( ( 2 ⁢ ν ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( 2 ⁢ ν ) ) + k + 1 ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence superscript 𝑝 2 0 formulae-sequence 2 𝜈 𝑘 1 0 2 𝜈 𝑘 1 0 {\displaystyle{\displaystyle|\Re\nu|<\tfrac{1}{2},\Re\left(p^{2}\right)>0,\Re(% (2\nu)+k+1)>0,\Re((-(2\nu))+k+1)>0}} 
int(BesselY(2*nu, b*t)*exp(- (p)^(2)* (t)^(2)), t = 0..infinity) = -(sqrt(Pi))/(2*p)*exp(-((b)^(2))/(8*(p)^(2)))*(BesselI(nu, ((b)^(2))/(8*(p)^(2)))*tan(nu*Pi)+(1)/(Pi)*BesselK(nu, ((b)^(2))/(8*(p)^(2)))*sec(nu*Pi))

Integrate[BesselY[2*\[Nu], b*t]*Exp[- (p)^(2)* (t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == -Divide[Sqrt[Pi],2*p]*Exp[-Divide[(b)^(2),8*(p)^(2)]]*(BesselI[\[Nu], Divide[(b)^(2),8*(p)^(2)]]*Tan[\[Nu]*Pi]+Divide[1,Pi]*BesselK[\[Nu], Divide[(b)^(2),8*(p)^(2)]]*Sec[\[Nu]*Pi])
Error Aborted - Skipped - Because timed out
10.22.E54 ∫ 0 ∞ J ν ⁡ ( b ⁢ t ) ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ t μ - 1 ⁢ d t = ( 1 2 ⁢ b / p ) ν ⁢ Γ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ ) 2 ⁢ p μ ⁢ exp ⁡ ( - b 2 4 ⁢ p 2 ) ⁢ 𝐌 ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ μ + 1 , ν + 1 , b 2 4 ⁢ p 2 ) superscript subscript 0 Bessel-J 𝜈 𝑏 𝑡 superscript 𝑝 2 superscript 𝑡 2 superscript 𝑡 𝜇 1 𝑡 superscript 1 2 𝑏 𝑝 𝜈 Euler-Gamma 1 2 𝜈 1 2 𝜇 2 superscript 𝑝 𝜇 superscript 𝑏 2 4 superscript 𝑝 2 Kummer-confluent-hypergeometric-bold-M 1 2 𝜈 1 2 𝜇 1 𝜈 1 superscript 𝑏 2 4 superscript 𝑝 2 {\displaystyle{\displaystyle\int_{0}^{\infty}J_{\nu}\left(bt\right)\exp\left(-% p^{2}t^{2}\right)t^{\mu-1}\mathrm{d}t=\frac{(\tfrac{1}{2}b/p)^{\nu}\Gamma\left% (\tfrac{1}{2}\nu+\tfrac{1}{2}\mu\right)}{2p^{\mu}}\exp\left(-\frac{b^{2}}{4p^{% 2}}\right)\*{\mathbf{M}}\left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+1,\nu+1,\frac{b^% {2}}{4p^{2}}\right)}}
\int_{0}^{\infty}\BesselJ{\nu}@{bt}\exp@{-p^{2}t^{2}}t^{\mu-1}\diff{t} = \frac{(\tfrac{1}{2}b/p)^{\nu}\EulerGamma@{\tfrac{1}{2}\nu+\tfrac{1}{2}\mu}}{2p^{\mu}}\exp@{-\frac{b^{2}}{4p^{2}}}\*\OlverconfhyperM@{\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+1}{\nu+1}{\frac{b^{2}}{4p^{2}}}		 ℜ ⁡ ( μ + ν ) > 0 , ℜ ⁡ ( p 2 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ ) > 0 formulae-sequence 𝜇 𝜈 0 formulae-sequence superscript 𝑝 2 0 formulae-sequence 𝜈 𝑘 1 0 1 2 𝜈 1 2 𝜇 0 {\displaystyle{\displaystyle\Re\left(\mu+\nu\right)>0,\Re\left(p^{2}\right)>0,% \Re(\nu+k+1)>0,\Re(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu)>0}} 
int(BesselJ(nu, b*t)*exp(- (p)^(2)* (t)^(2))*(t)^(mu - 1), t = 0..infinity) = (((1)/(2)*b/p)^(nu)* GAMMA((1)/(2)*nu +(1)/(2)*mu))/(2*(p)^(mu))*exp(-((b)^(2))/(4*(p)^(2)))* KummerM((1)/(2)*nu -(1)/(2)*mu + 1, nu + 1, ((b)^(2))/(4*(p)^(2)))/GAMMA(nu + 1)

Integrate[BesselJ[\[Nu], b*t]*Exp[- (p)^(2)* (t)^(2)]*(t)^(\[Mu]- 1), {t, 0, Infinity}, GenerateConditions->None] == Divide[(Divide[1,2]*b/p)^\[Nu]* Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]],2*(p)^\[Mu]]*Exp[-Divide[(b)^(2),4*(p)^(2)]]* Hypergeometric1F1Regularized[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+ 1, \[Nu]+ 1, Divide[(b)^(2),4*(p)^(2)]]
Error Aborted -
Failed [246 / 300]
Result: Complex[0.07541885663346475, -0.6281916024632631]
Test Values: {Rule[b, -1.5], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.1002850405400357, -0.7734416454563844]
Test Values: {Rule[b, -1.5], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.22.E55 ∫ 0 ∞ t - 1 ⁢ J ν + 2 ⁢ ℓ + 1 ⁡ ( t ) ⁢ J ν + 2 ⁢ m + 1 ⁡ ( t ) ⁢ d t = δ ℓ , m 2 ⁢ ( 2 ⁢ ℓ + ν + 1 ) superscript subscript 0 superscript 𝑡 1 Bessel-J 𝜈 2 ℓ 1 𝑡 Bessel-J 𝜈 2 𝑚 1 𝑡 𝑡 Kronecker ℓ 𝑚 2 2 ℓ 𝜈 1 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{-1}J_{\nu+2\ell+1}\left(t% \right)J_{\nu+2m+1}\left(t\right)\mathrm{d}t=\frac{\delta_{\ell,m}}{2(2\ell+% \nu+1)}}}
\int_{0}^{\infty}t^{-1}\BesselJ{\nu+2\ell+1}@{t}\BesselJ{\nu+2m+1}@{t}\diff{t} = \frac{\Kroneckerdelta{\ell}{m}}{2(2\ell+\nu+1)}		 ν + ℓ + m > - 1 , ℜ ⁡ ( ( ν + 2 ⁢ ℓ + 1 ) + k + 1 ) > 0 , ℜ ⁡ ( ( ν + 2 ⁢ m + 1 ) + k + 1 ) > 0 formulae-sequence 𝜈 ℓ 𝑚 1 formulae-sequence 𝜈 2 ℓ 1 𝑘 1 0 𝜈 2 𝑚 1 𝑘 1 0 {\displaystyle{\displaystyle\nu+\ell+m>-1,\Re((\nu+2\ell+1)+k+1)>0,\Re((\nu+2m% +1)+k+1)>0}} 
int((t)^(- 1)* BesselJ(nu + 2*ell + 1, t)*BesselJ(nu + 2*m + 1, t), t = 0..infinity) = (KroneckerDelta[ell, m])/(2*(2*ell + nu + 1))

Integrate[(t)^(- 1)* BesselJ[\[Nu]+ 2*\[ScriptL]+ 1, t]*BesselJ[\[Nu]+ 2*m + 1, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[KroneckerDelta[\[ScriptL], m],2*(2*\[ScriptL]+ \[Nu]+ 1)]
Failure Failure Error
Failed [18 / 54]
Result: Indeterminate
Test Values: {Rule[m, 1], Rule[ℓ, 1], Rule[ν, Rational[-3, 2]]}


Result: Indeterminate
Test Values: {Rule[m, 2], Rule[ℓ, 2], Rule[ν, Rational[-3, 2]]}

... skip entries to safe data
10.22.E56 ∫ 0 ∞ J μ ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( b ⁢ t ) t λ ⁢ d t = a μ ⁢ Γ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ - 1 2 ⁢ λ + 1 2 ) 2 λ ⁢ b μ - λ + 1 ⁢ Γ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ μ + 1 2 ⁢ λ + 1 2 ) ⁢ 𝐅 ⁡ ( 1 2 ⁢ ( μ + ν - λ + 1 ) , 1 2 ⁢ ( μ - ν - λ + 1 ) ; μ + 1 ; a 2 b 2 ) superscript subscript 0 Bessel-J 𝜇 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 superscript 𝑡 𝜆 𝑡 superscript 𝑎 𝜇 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 2 𝜆 1 2 superscript 2 𝜆 superscript 𝑏 𝜇 𝜆 1 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 2 𝜆 1 2 scaled-hypergeometric-bold-F 1 2 𝜇 𝜈 𝜆 1 1 2 𝜇 𝜈 𝜆 1 𝜇 1 superscript 𝑎 2 superscript 𝑏 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{J_{\mu}\left(at\right)J_{% \nu}\left(bt\right)}{t^{\lambda}}\mathrm{d}t=\frac{a^{\mu}\Gamma\left(\frac{1}% {2}\nu+\frac{1}{2}\mu-\frac{1}{2}\lambda+\frac{1}{2}\right)}{2^{\lambda}b^{\mu% -\lambda+1}\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\lambda+\frac{% 1}{2}\right)}\*\mathbf{F}\left(\tfrac{1}{2}(\mu+\nu-\lambda+1),\tfrac{1}{2}(% \mu-\nu-\lambda+1);\mu+1;\frac{a^{2}}{b^{2}}\right)}}
\int_{0}^{\infty}\frac{\BesselJ{\mu}@{at}\BesselJ{\nu}@{bt}}{t^{\lambda}}\diff{t} = \frac{a^{\mu}\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu-\frac{1}{2}\lambda+\frac{1}{2}}}{2^{\lambda}b^{\mu-\lambda+1}\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\lambda+\frac{1}{2}}}\*\hyperOlverF@{\tfrac{1}{2}(\mu+\nu-\lambda+1)}{\tfrac{1}{2}(\mu-\nu-\lambda+1)}{\mu+1}{\frac{a^{2}}{b^{2}}}		 0 < a , a < b , ℜ ⁡ ( μ + ν + 1 ) > ℜ ⁡ λ , ℜ ⁡ λ > - 1 , ℜ ⁡ ( ( μ ) + k + 1 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ - 1 2 ⁢ λ + 1 2 ) > 0 , ℜ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ μ + 1 2 ⁢ λ + 1 2 ) > 0 formulae-sequence 0 𝑎 formulae-sequence 𝑎 𝑏 formulae-sequence 𝜇 𝜈 1 𝜆 formulae-sequence 𝜆 1 formulae-sequence 𝜇 𝑘 1 0 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 1 2 𝜈 1 2 𝜇 1 2 𝜆 1 2 0 1 2 𝜈 1 2 𝜇 1 2 𝜆 1 2 0 {\displaystyle{\displaystyle 0<a,a<b,\Re\left(\mu+\nu+1\right)>\Re\lambda,\Re% \lambda>-1,\Re((\mu)+k+1)>0,\Re(\nu+k+1)>0,\Re(\frac{1}{2}\nu+\frac{1}{2}\mu-% \frac{1}{2}\lambda+\frac{1}{2})>0,\Re(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2% }\lambda+\frac{1}{2})>0}} 
int((BesselJ(mu, a*t)*BesselJ(nu, b*t))/((t)^(lambda)), t = 0..infinity) = ((a)^(mu)* GAMMA((1)/(2)*nu +(1)/(2)*mu -(1)/(2)*lambda +(1)/(2)))/((2)^(lambda)* (b)^(mu - lambda + 1)* GAMMA((1)/(2)*nu -(1)/(2)*mu +(1)/(2)*lambda +(1)/(2)))* hypergeom([(1)/(2)*(mu + nu - lambda + 1), (1)/(2)*(mu - nu - lambda + 1)], [mu + 1], ((a)^(2))/((b)^(2)))/GAMMA(mu + 1)

Integrate[Divide[BesselJ[\[Mu], a*t]*BesselJ[\[Nu], b*t],(t)^\[Lambda]], {t, 0, Infinity}, GenerateConditions->None] == Divide[(a)^\[Mu]* Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]-Divide[1,2]*\[Lambda]+Divide[1,2]],(2)^\[Lambda]* (b)^(\[Mu]- \[Lambda]+ 1)* Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2]*\[Lambda]+Divide[1,2]]]* Hypergeometric2F1Regularized[Divide[1,2]*(\[Mu]+ \[Nu]- \[Lambda]+ 1), Divide[1,2]*(\[Mu]- \[Nu]- \[Lambda]+ 1), \[Mu]+ 1, Divide[(a)^(2),(b)^(2)]]
Error Aborted -
Failed [300 / 300]
Result: Complex[0.12507202091813296, -0.11002587193353452]
Test Values: {Rule[a, 1.5], Rule[b, 2], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.017959797138118128, 0.3252875517547388]
Test Values: {Rule[a, 1.5], Rule[b, 2], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.22.E57 ∫ 0 ∞ J μ ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( a ⁢ t ) t λ ⁢ d t = ( 1 2 ⁢ a ) λ - 1 ⁢ Γ ⁡ ( 1 2 ⁢ μ + 1 2 ⁢ ν - 1 2 ⁢ λ + 1 2 ) ⁢ Γ ⁡ ( λ ) 2 ⁢ Γ ⁡ ( 1 2 ⁢ λ + 1 2 ⁢ ν - 1 2 ⁢ μ + 1 2 ) ⁢ Γ ⁡ ( 1 2 ⁢ λ + 1 2 ⁢ μ - 1 2 ⁢ ν + 1 2 ) ⁢ Γ ⁡ ( 1 2 ⁢ λ + 1 2 ⁢ μ + 1 2 ⁢ ν + 1 2 ) superscript subscript 0 Bessel-J 𝜇 𝑎 𝑡 Bessel-J 𝜈 𝑎 𝑡 superscript 𝑡 𝜆 𝑡 superscript 1 2 𝑎 𝜆 1 Euler-Gamma 1 2 𝜇 1 2 𝜈 1 2 𝜆 1 2 Euler-Gamma 𝜆 2 Euler-Gamma 1 2 𝜆 1 2 𝜈 1 2 𝜇 1 2 Euler-Gamma 1 2 𝜆 1 2 𝜇 1 2 𝜈 1 2 Euler-Gamma 1 2 𝜆 1 2 𝜇 1 2 𝜈 1 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{J_{\mu}\left(at\right)J_{% \nu}\left(at\right)}{t^{\lambda}}\mathrm{d}t=\frac{(\frac{1}{2}a)^{\lambda-1}% \Gamma\left(\frac{1}{2}\mu+\frac{1}{2}\nu-\frac{1}{2}\lambda+\frac{1}{2}\right% )\Gamma\left(\lambda\right)}{2\Gamma\left(\frac{1}{2}\lambda+\frac{1}{2}\nu-% \frac{1}{2}\mu+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\lambda+\frac{1}{2}\mu% -\frac{1}{2}\nu+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\lambda+\frac{1}{2}% \mu+\frac{1}{2}\nu+\frac{1}{2}\right)}}}
\int_{0}^{\infty}\frac{\BesselJ{\mu}@{at}\BesselJ{\nu}@{at}}{t^{\lambda}}\diff{t} = \frac{(\frac{1}{2}a)^{\lambda-1}\EulerGamma@{\frac{1}{2}\mu+\frac{1}{2}\nu-\frac{1}{2}\lambda+\frac{1}{2}}\EulerGamma@{\lambda}}{2\EulerGamma@{\frac{1}{2}\lambda+\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}}\EulerGamma@{\frac{1}{2}\lambda+\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2}}\EulerGamma@{\frac{1}{2}\lambda+\frac{1}{2}\mu+\frac{1}{2}\nu+\frac{1}{2}}}		 ℜ ⁡ ( μ + ν + 1 ) > ℜ ⁡ λ , ℜ ⁡ λ > 0 , ℜ ⁡ ( ( μ ) + k + 1 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( 1 2 ⁢ μ + 1 2 ⁢ ν - 1 2 ⁢ λ + 1 2 ) > 0 , ℜ ⁡ ( λ ) > 0 , ℜ ⁡ ( 1 2 ⁢ λ + 1 2 ⁢ ν - 1 2 ⁢ μ + 1 2 ) > 0 , ℜ ⁡ ( 1 2 ⁢ λ + 1 2 ⁢ μ - 1 2 ⁢ ν + 1 2 ) > 0 , ℜ ⁡ ( 1 2 ⁢ λ + 1 2 ⁢ μ + 1 2 ⁢ ν + 1 2 ) > 0 formulae-sequence 𝜇 𝜈 1 𝜆 formulae-sequence 𝜆 0 formulae-sequence 𝜇 𝑘 1 0 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 1 2 𝜇 1 2 𝜈 1 2 𝜆 1 2 0 formulae-sequence 𝜆 0 formulae-sequence 1 2 𝜆 1 2 𝜈 1 2 𝜇 1 2 0 formulae-sequence 1 2 𝜆 1 2 𝜇 1 2 𝜈 1 2 0 1 2 𝜆 1 2 𝜇 1 2 𝜈 1 2 0 {\displaystyle{\displaystyle\Re\left(\mu+\nu+1\right)>\Re\lambda,\Re\lambda>0,% \Re((\mu)+k+1)>0,\Re(\nu+k+1)>0,\Re(\frac{1}{2}\mu+\frac{1}{2}\nu-\frac{1}{2}% \lambda+\frac{1}{2})>0,\Re(\lambda)>0,\Re(\frac{1}{2}\lambda+\frac{1}{2}\nu-% \frac{1}{2}\mu+\frac{1}{2})>0,\Re(\frac{1}{2}\lambda+\frac{1}{2}\mu-\frac{1}{2% }\nu+\frac{1}{2})>0,\Re(\frac{1}{2}\lambda+\frac{1}{2}\mu+\frac{1}{2}\nu+\frac% {1}{2})>0}} 
int((BesselJ(mu, a*t)*BesselJ(nu, a*t))/((t)^(lambda)), t = 0..infinity) = (((1)/(2)*a)^(lambda - 1)* GAMMA((1)/(2)*mu +(1)/(2)*nu -(1)/(2)*lambda +(1)/(2))*GAMMA(lambda))/(2*GAMMA((1)/(2)*lambda +(1)/(2)*nu -(1)/(2)*mu +(1)/(2))*GAMMA((1)/(2)*lambda +(1)/(2)*mu -(1)/(2)*nu +(1)/(2))*GAMMA((1)/(2)*lambda +(1)/(2)*mu +(1)/(2)*nu +(1)/(2)))

Integrate[Divide[BesselJ[\[Mu], a*t]*BesselJ[\[Nu], a*t],(t)^\[Lambda]], {t, 0, Infinity}, GenerateConditions->None] == Divide[(Divide[1,2]*a)^(\[Lambda]- 1)* Gamma[Divide[1,2]*\[Mu]+Divide[1,2]*\[Nu]-Divide[1,2]*\[Lambda]+Divide[1,2]]*Gamma[\[Lambda]],2*Gamma[Divide[1,2]*\[Lambda]+Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2]]*Gamma[Divide[1,2]*\[Lambda]+Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]+Divide[1,2]]*Gamma[Divide[1,2]*\[Lambda]+Divide[1,2]*\[Mu]+Divide[1,2]*\[Nu]+Divide[1,2]]]
Error Aborted - Skipped - Because timed out
10.22.E58 ∫ 0 ∞ J ν ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( b ⁢ t ) t λ ⁢ d t = ( a ⁢ b ) ν ⁢ Γ ⁡ ( ν - 1 2 ⁢ λ + 1 2 ) 2 λ ⁢ ( a 2 + b 2 ) ν - 1 2 ⁢ λ + 1 2 ⁢ Γ ⁡ ( 1 2 ⁢ λ + 1 2 ) ⁢ 𝐅 ⁡ ( 2 ⁢ ν + 1 - λ 4 , 2 ⁢ ν + 3 - λ 4 ; ν + 1 ; 4 ⁢ a 2 ⁢ b 2 ( a 2 + b 2 ) 2 ) superscript subscript 0 Bessel-J 𝜈 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 superscript 𝑡 𝜆 𝑡 superscript 𝑎 𝑏 𝜈 Euler-Gamma 𝜈 1 2 𝜆 1 2 superscript 2 𝜆 superscript superscript 𝑎 2 superscript 𝑏 2 𝜈 1 2 𝜆 1 2 Euler-Gamma 1 2 𝜆 1 2 scaled-hypergeometric-bold-F 2 𝜈 1 𝜆 4 2 𝜈 3 𝜆 4 𝜈 1 4 superscript 𝑎 2 superscript 𝑏 2 superscript superscript 𝑎 2 superscript 𝑏 2 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{J_{\nu}\left(at\right)J_{% \nu}\left(bt\right)}{t^{\lambda}}\mathrm{d}t=\frac{(ab)^{\nu}\Gamma\left(\nu-% \frac{1}{2}\lambda+\frac{1}{2}\right)}{2^{\lambda}(a^{2}+b^{2})^{\nu-\frac{1}{% 2}\lambda+\frac{1}{2}}\Gamma\left(\frac{1}{2}\lambda+\frac{1}{2}\right)}% \mathbf{F}\left(\frac{2\nu+1-\lambda}{4},\frac{2\nu+3-\lambda}{4};\nu+1;\frac{% 4a^{2}b^{2}}{(a^{2}+b^{2})^{2}}\right)}}
\int_{0}^{\infty}\frac{\BesselJ{\nu}@{at}\BesselJ{\nu}@{bt}}{t^{\lambda}}\diff{t} = \frac{(ab)^{\nu}\EulerGamma@{\nu-\frac{1}{2}\lambda+\frac{1}{2}}}{2^{\lambda}(a^{2}+b^{2})^{\nu-\frac{1}{2}\lambda+\frac{1}{2}}\EulerGamma@{\frac{1}{2}\lambda+\frac{1}{2}}}\hyperOlverF@{\frac{2\nu+1-\lambda}{4}}{\frac{2\nu+3-\lambda}{4}}{\nu+1}{\frac{4a^{2}b^{2}}{(a^{2}+b^{2})^{2}}}		 a ≠ b , ℜ ⁡ ( 2 ⁢ ν + 1 ) > ℜ ⁡ λ , ℜ ⁡ λ > - 1 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ν - 1 2 ⁢ λ + 1 2 ) > 0 , ℜ ⁡ ( 1 2 ⁢ λ + 1 2 ) > 0 formulae-sequence 𝑎 𝑏 formulae-sequence 2 𝜈 1 𝜆 formulae-sequence 𝜆 1 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 𝜈 1 2 𝜆 1 2 0 1 2 𝜆 1 2 0 {\displaystyle{\displaystyle a\neq b,\Re\left(2\nu+1\right)>\Re\lambda,\Re% \lambda>-1,\Re(\nu+k+1)>0,\Re(\nu-\frac{1}{2}\lambda+\frac{1}{2})>0,\Re(\frac{% 1}{2}\lambda+\frac{1}{2})>0}} 
int((BesselJ(nu, a*t)*BesselJ(nu, b*t))/((t)^(lambda)), t = 0..infinity) = ((a*b)^(nu)* GAMMA(nu -(1)/(2)*lambda +(1)/(2)))/((2)^(lambda)*((a)^(2)+ (b)^(2))^(nu -(1)/(2)*lambda +(1)/(2))* GAMMA((1)/(2)*lambda +(1)/(2)))*hypergeom([(2*nu + 1 - lambda)/(4), (2*nu + 3 - lambda)/(4)], [nu + 1], (4*(a)^(2)* (b)^(2))/(((a)^(2)+ (b)^(2))^(2)))/GAMMA(nu + 1)

Integrate[Divide[BesselJ[\[Nu], a*t]*BesselJ[\[Nu], b*t],(t)^\[Lambda]], {t, 0, Infinity}, GenerateConditions->None] == Divide[(a*b)^\[Nu]* Gamma[\[Nu]-Divide[1,2]*\[Lambda]+Divide[1,2]],(2)^\[Lambda]*((a)^(2)+ (b)^(2))^(\[Nu]-Divide[1,2]*\[Lambda]+Divide[1,2])* Gamma[Divide[1,2]*\[Lambda]+Divide[1,2]]]*Hypergeometric2F1Regularized[Divide[2*\[Nu]+ 1 - \[Lambda],4], Divide[2*\[Nu]+ 3 - \[Lambda],4], \[Nu]+ 1, Divide[4*(a)^(2)* (b)^(2),((a)^(2)+ (b)^(2))^(2)]]
Error Aborted -
Failed [209 / 300]
Result: Complex[-0.13393539357334844, 0.1322614378889556]
Test Values: {Rule[a, -1.5], Rule[b, -0.5], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.07230690300251369, -0.15068591568973605]
Test Values: {Rule[a, -1.5], Rule[b, -0.5], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
10.22.E66 ∫ 0 ∞ e - a ⁢ t ⁢ J ν ⁡ ( b ⁢ t ) ⁢ J ν ⁡ ( c ⁢ t ) ⁢ d t = 1 π ⁢ ( b ⁢ c ) 1 2 ⁢ Q ν - 1 2 ⁡ ( a 2 + b 2 + c 2 2 ⁢ b ⁢ c ) superscript subscript 0 superscript 𝑒 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 Bessel-J 𝜈 𝑐 𝑡 𝑡 1 𝜋 superscript 𝑏 𝑐 1 2 shorthand-Legendre-Q-second-kind 𝜈 1 2 superscript 𝑎 2 superscript 𝑏 2 superscript 𝑐 2 2 𝑏 𝑐 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}J_{\nu}\left(bt\right)J_{% \nu}\left(ct\right)\mathrm{d}t=\frac{1}{\pi(bc)^{\frac{1}{2}}}\*Q_{\nu-\frac{1% }{2}}\left(\frac{a^{2}+b^{2}+c^{2}}{2bc}\right)}}
\int_{0}^{\infty}e^{-at}\BesselJ{\nu}@{bt}\BesselJ{\nu}@{ct}\diff{t} = \frac{1}{\pi(bc)^{\frac{1}{2}}}\*\assLegendreQ[]{\nu-\frac{1}{2}}@{\frac{a^{2}+b^{2}+c^{2}}{2bc}}		 ℜ ⁡ ν > - 1 2 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 1 2 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+k+1)>0}} 
int(exp(- a*t)*BesselJ(nu, b*t)*BesselJ(nu, c*t), t = 0..infinity) = (1)/(Pi*(b*c)^((1)/(2)))* LegendreQ(nu -(1)/(2), ((a)^(2)+ (b)^(2)+ (c)^(2))/(2*b*c))

Integrate[Exp[- a*t]*BesselJ[\[Nu], b*t]*BesselJ[\[Nu], c*t], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,Pi*(b*c)^(Divide[1,2])]* LegendreQ[\[Nu]-Divide[1,2], 0, 3, Divide[(a)^(2)+ (b)^(2)+ (c)^(2),2*b*c]]
Error Aborted - Skipped - Because timed out
10.22.E67 ∫ 0 ∞ t ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ J ν ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( b ⁢ t ) ⁢ d t = 1 2 ⁢ p 2 ⁢ exp ⁡ ( - a 2 + b 2 4 ⁢ p 2 ) ⁢ I ν ⁡ ( a ⁢ b 2 ⁢ p 2 ) superscript subscript 0 𝑡 superscript 𝑝 2 superscript 𝑡 2 Bessel-J 𝜈 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 𝑡 1 2 superscript 𝑝 2 superscript 𝑎 2 superscript 𝑏 2 4 superscript 𝑝 2 modified-Bessel-first-kind 𝜈 𝑎 𝑏 2 superscript 𝑝 2 {\displaystyle{\displaystyle\int_{0}^{\infty}t\exp\left(-p^{2}t^{2}\right)J_{% \nu}\left(at\right)J_{\nu}\left(bt\right)\mathrm{d}t=\frac{1}{2p^{2}}\exp\left% (-\frac{a^{2}+b^{2}}{4p^{2}}\right)I_{\nu}\left(\frac{ab}{2p^{2}}\right)}}
\int_{0}^{\infty}t\exp@{-p^{2}t^{2}}\BesselJ{\nu}@{at}\BesselJ{\nu}@{bt}\diff{t} = \frac{1}{2p^{2}}\exp@{-\frac{a^{2}+b^{2}}{4p^{2}}}\modBesselI{\nu}\left(\frac{ab}{2p^{2}}\right)		 ℜ ⁡ ν > - 1 , ℜ ⁡ ( p 2 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 1 formulae-sequence superscript 𝑝 2 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\nu>-1,\Re\left(p^{2}\right)>0,\Re(\nu+k+1)>0}} 
int(t*exp(- (p)^(2)* (t)^(2))*BesselJ(nu, a*t)*BesselJ(nu, b*t), t = 0..infinity) = (1)/(2*(p)^(2))*exp(-((a)^(2)+ (b)^(2))/(4*(p)^(2)))*BesselI(nu, (a*b)/(2*(p)^(2)))

Integrate[t*Exp[- (p)^(2)* (t)^(2)]*BesselJ[\[Nu], a*t]*BesselJ[\[Nu], b*t], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,2*(p)^(2)]*Exp[-Divide[(a)^(2)+ (b)^(2),4*(p)^(2)]]*BesselI[\[Nu], Divide[a*b,2*(p)^(2)]]
Translation Error Translation Error - -
10.22.E68 ∫ 0 ∞ t ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ J 0 ⁡ ( a ⁢ t ) ⁢ Y 0 ⁡ ( a ⁢ t ) ⁢ d t = - 1 2 ⁢ π ⁢ p 2 ⁢ exp ⁡ ( - a 2 2 ⁢ p 2 ) ⁢ K 0 ⁡ ( a 2 2 ⁢ p 2 ) superscript subscript 0 𝑡 superscript 𝑝 2 superscript 𝑡 2 Bessel-J 0 𝑎 𝑡 Bessel-Y-Weber 0 𝑎 𝑡 𝑡 1 2 𝜋 superscript 𝑝 2 superscript 𝑎 2 2 superscript 𝑝 2 modified-Bessel-second-kind 0 superscript 𝑎 2 2 superscript 𝑝 2 {\displaystyle{\displaystyle\int_{0}^{\infty}t\exp\left(-p^{2}t^{2}\right)J_{0% }\left(at\right)Y_{0}\left(at\right)\mathrm{d}t=-\frac{1}{2\pi p^{2}}\exp\left% (-\frac{a^{2}}{2p^{2}}\right)K_{0}\left(\frac{a^{2}}{2p^{2}}\right)}}
\int_{0}^{\infty}t\exp@{-p^{2}t^{2}}\BesselJ{0}@{at}\BesselY{0}@{at}\diff{t} = -\frac{1}{2\pi p^{2}}\exp@{-\frac{a^{2}}{2p^{2}}}\modBesselK{0}\left(\frac{a^{2}}{2p^{2}}\right)		 ℜ ⁡ ( p 2 ) > 0 , ℜ ⁡ ( 0 + k + 1 ) > 0 , ℜ ⁡ ( ( - 0 ) + k + 1 ) > 0 formulae-sequence superscript 𝑝 2 0 formulae-sequence 0 𝑘 1 0 0 𝑘 1 0 {\displaystyle{\displaystyle\Re\left(p^{2}\right)>0,\Re(0+k+1)>0,\Re((-0)+k+1)% >0}} 
int(t*exp(- (p)^(2)* (t)^(2))*BesselJ(0, a*t)*BesselY(0, a*t), t = 0..infinity) = -(1)/(2*Pi*(p)^(2))*exp(-((a)^(2))/(2*(p)^(2)))*BesselK(0, ((a)^(2))/(2*(p)^(2)))

Integrate[t*Exp[- (p)^(2)* (t)^(2)]*BesselJ[0, a*t]*BesselY[0, a*t], {t, 0, Infinity}, GenerateConditions->None] == -Divide[1,2*Pi*(p)^(2)]*Exp[-Divide[(a)^(2),2*(p)^(2)]]*BesselK[0, Divide[(a)^(2),2*(p)^(2)]]
Translation Error Translation Error - -
10.22.E70 ∫ 0 ∞ Y ν ⁡ ( a ⁢ t ) ⁢ J ν + 1 ⁡ ( b ⁢ t ) ⁢ t ⁢ d t t 2 - z 2 = 1 2 ⁢ π ⁢ J ν + 1 ⁡ ( b ⁢ z ) ⁢ H ν ( 1 ) ⁡ ( a ⁢ z ) superscript subscript 0 Bessel-Y-Weber 𝜈 𝑎 𝑡 Bessel-J 𝜈 1 𝑏 𝑡 𝑡 𝑡 superscript 𝑡 2 superscript 𝑧 2 1 2 𝜋 Bessel-J 𝜈 1 𝑏 𝑧 Hankel-H-1-Bessel-third-kind 𝜈 𝑎 𝑧 {\displaystyle{\displaystyle\int_{0}^{\infty}Y_{\nu}\left(at\right)J_{\nu+1}% \left(bt\right)\frac{t\mathrm{d}t}{t^{2}-z^{2}}=\frac{1}{2}\pi J_{\nu+1}\left(% bz\right){H^{(1)}_{\nu}}\left(az\right)}}
\int_{0}^{\infty}\BesselY{\nu}@{at}\BesselJ{\nu+1}@{bt}\frac{t\diff{t}}{t^{2}-z^{2}} = \frac{1}{2}\pi\BesselJ{\nu+1}@{bz}\HankelH{1}{\nu}@{az}		 a ≥ b , b > 0 , ℜ ⁡ ν > - 3 2 , ℑ ⁡ z > 0 , ℜ ⁡ ( ( ν + 1 ) + k + 1 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 formulae-sequence 𝑎 𝑏 formulae-sequence 𝑏 0 formulae-sequence 𝜈 3 2 formulae-sequence 𝑧 0 formulae-sequence 𝜈 1 𝑘 1 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle a\geq b,b>0,\Re\nu>-\tfrac{3}{2},\Im z>0,\Re((\nu% +1)+k+1)>0,\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}} 
int(BesselY(nu, a*t)*BesselJ(nu + 1, b*t)*(t)/((t)^(2)- (z)^(2)), t = 0..infinity) = (1)/(2)*Pi*BesselJ(nu + 1, b*z)*HankelH1(nu, a*z)

Integrate[BesselY[\[Nu], a*t]*BesselJ[\[Nu]+ 1, b*t]*Divide[t,(t)^(2)- (z)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,2]*Pi*BesselJ[\[Nu]+ 1, b*z]*HankelH1[\[Nu], a*z]
Error Aborted - Skipped - Because timed out
10.22.E71 ∫ 0 ∞ J μ ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( b ⁢ t ) ⁢ J ν ⁡ ( c ⁢ t ) ⁢ t 1 - μ ⁢ d t = ( b ⁢ c ) μ - 1 ⁢ ( sin ⁡ ϕ ) μ - 1 2 ( 2 ⁢ π ) 1 2 ⁢ a μ ⁢ 𝖯 ν - 1 2 1 2 - μ ⁡ ( cos ⁡ ϕ ) superscript subscript 0 Bessel-J 𝜇 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 Bessel-J 𝜈 𝑐 𝑡 superscript 𝑡 1 𝜇 𝑡 superscript 𝑏 𝑐 𝜇 1 superscript italic-ϕ 𝜇 1 2 superscript 2 𝜋 1 2 superscript 𝑎 𝜇 Ferrers-Legendre-P-first-kind 1 2 𝜇 𝜈 1 2 italic-ϕ {\displaystyle{\displaystyle\int_{0}^{\infty}J_{\mu}\left(at\right)J_{\nu}% \left(bt\right)J_{\nu}\left(ct\right)t^{1-\mu}\mathrm{d}t=\frac{(bc)^{\mu-1}(% \sin\phi)^{\mu-\frac{1}{2}}}{(2\pi)^{\frac{1}{2}}a^{\mu}}\mathsf{P}^{\frac{1}{% 2}-\mu}_{\nu-\frac{1}{2}}(\cos\phi)}}
\int_{0}^{\infty}\BesselJ{\mu}@{at}\BesselJ{\nu}@{bt}\BesselJ{\nu}@{ct}t^{1-\mu}\diff{t} = \frac{(bc)^{\mu-1}(\sin@@{\phi})^{\mu-\frac{1}{2}}}{(2\pi)^{\frac{1}{2}}a^{\mu}}\FerrersP[\frac{1}{2}-\mu]{\nu-\frac{1}{2}}(\cos@@{\phi})		 ℜ ⁡ μ > - 1 2 , ℜ ⁡ ν > - 1 , | b - c | < a , a < b + c , cos ⁡ ϕ = ( b 2 + c 2 - a 2 ) / ( 2 ⁢ b ⁢ c ) , ℜ ⁡ ( ( μ ) + k + 1 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜇 1 2 formulae-sequence 𝜈 1 formulae-sequence 𝑏 𝑐 𝑎 formulae-sequence 𝑎 𝑏 𝑐 formulae-sequence italic-ϕ superscript 𝑏 2 superscript 𝑐 2 superscript 𝑎 2 2 𝑏 𝑐 formulae-sequence 𝜇 𝑘 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2},\Re\nu>-1,|b-c|<a,a<b+c,\cos% \phi=(b^{2}+c^{2}-a^{2})/(2bc),\Re((\mu)+k+1)>0,\Re(\nu+k+1)>0}} 
int(BesselJ(mu, a*t)*BesselJ(nu, b*t)*BesselJ(nu, c*t)*(t)^(1 - mu), t = 0..infinity) = ((b*c)^(mu - 1)*(sin(phi))^(mu -(1)/(2)))/((2*Pi)^((1)/(2))* (a)^(mu))*LegendreP(nu -(1)/(2), (1)/(2)- mu, cos(phi))

Integrate[BesselJ[\[Mu], a*t]*BesselJ[\[Nu], b*t]*BesselJ[\[Nu], c*t]*(t)^(1 - \[Mu]), {t, 0, Infinity}, GenerateConditions->None] == Divide[(b*c)^(\[Mu]- 1)*(Sin[\[Phi]])^(\[Mu]-Divide[1,2]),(2*Pi)^(Divide[1,2])* (a)^\[Mu]]*LegendreP[\[Nu]-Divide[1,2], Divide[1,2]- \[Mu], Cos[\[Phi]]]
Translation Error Translation Error - -
10.22.E72 ∫ 0 ∞ J μ ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( b ⁢ t ) ⁢ J ν ⁡ ( c ⁢ t ) ⁢ t 1 - μ ⁢ d t = ( b ⁢ c ) μ - 1 ⁢ sin ⁡ ( ( μ - ν ) ⁢ π ) ⁢ ( sinh ⁡ χ ) μ - 1 2 ( 1 2 ⁢ π 3 ) 1 2 ⁢ a μ ⁢ e ( μ - 1 2 ) ⁢ i ⁢ π ⁢ Q ν - 1 2 1 2 - μ ⁡ ( cosh ⁡ χ ) superscript subscript 0 Bessel-J 𝜇 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 Bessel-J 𝜈 𝑐 𝑡 superscript 𝑡 1 𝜇 𝑡 superscript 𝑏 𝑐 𝜇 1 𝜇 𝜈 superscript 𝜒 𝜇 1 2 superscript 1 2 superscript 𝜋 3 1 2 superscript 𝑎 𝜇 𝜇 1 2 imaginary-unit Legendre-Q-second-kind 1 2 𝜇 𝜈 1 2 𝜒 {\displaystyle{\displaystyle\int_{0}^{\infty}J_{\mu}\left(at\right)J_{\nu}% \left(bt\right)J_{\nu}\left(ct\right)t^{1-\mu}\mathrm{d}t=\frac{(bc)^{\mu-1}% \sin\left((\mu-\nu)\pi\right)(\sinh\chi)^{\mu-\frac{1}{2}}}{(\frac{1}{2}\pi^{3% })^{\frac{1}{2}}a^{\mu}}{\mathrm{e}^{(\mu-\frac{1}{2})\mathrm{i}\pi}}Q^{\frac{% 1}{2}-\mu}_{\nu-\frac{1}{2}}\left(\cosh\chi\right)}}
\int_{0}^{\infty}\BesselJ{\mu}@{at}\BesselJ{\nu}@{bt}\BesselJ{\nu}@{ct}t^{1-\mu}\diff{t} = \frac{(bc)^{\mu-1}\sin@{(\mu-\nu)\cpi}(\sinh@@{\chi})^{\mu-\frac{1}{2}}}{(\frac{1}{2}\pi^{3})^{\frac{1}{2}}a^{\mu}}\expe^{(\mu-\frac{1}{2})\iunit\cpi}\assLegendreQ[\frac{1}{2}-\mu]{\nu-\frac{1}{2}}@{\cosh@@{\chi}}		 ℜ ⁡ μ > - 1 2 , ℜ ⁡ ν > - 1 , a > b + c , cosh ⁡ χ = ( a 2 - b 2 - c 2 ) / ( 2 ⁢ b ⁢ c ) , ℜ ⁡ ( ( μ ) + k + 1 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜇 1 2 formulae-sequence 𝜈 1 formulae-sequence 𝑎 𝑏 𝑐 formulae-sequence 𝜒 superscript 𝑎 2 superscript 𝑏 2 superscript 𝑐 2 2 𝑏 𝑐 formulae-sequence 𝜇 𝑘 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2},\Re\nu>-1,a>b+c,\cosh\chi=(a^% {2}-b^{2}-c^{2})/(2bc),\Re((\mu)+k+1)>0,\Re(\nu+k+1)>0}} 
int(BesselJ(mu, a*t)*BesselJ(nu, b*t)*BesselJ(nu, c*t)*(t)^(1 - mu), t = 0..infinity) = ((b*c)^(mu - 1)* sin((mu - nu)*Pi)*(sinh(chi))^(mu -(1)/(2)))/(((1)/(2)*(Pi)^(3))^((1)/(2))* (a)^(mu))*exp((mu -(1)/(2))*I*Pi)*LegendreQ(nu -(1)/(2), (1)/(2)- mu, cosh(chi))

Integrate[BesselJ[\[Mu], a*t]*BesselJ[\[Nu], b*t]*BesselJ[\[Nu], c*t]*(t)^(1 - \[Mu]), {t, 0, Infinity}, GenerateConditions->None] == Divide[(b*c)^(\[Mu]- 1)* Sin[(\[Mu]- \[Nu])*Pi]*(Sinh[\[Chi]])^(\[Mu]-Divide[1,2]),(Divide[1,2]*(Pi)^(3))^(Divide[1,2])* (a)^\[Mu]]*Exp[(\[Mu]-Divide[1,2])*I*Pi]*LegendreQ[\[Nu]-Divide[1,2], Divide[1,2]- \[Mu], 3, Cosh[\[Chi]]]
Error Aborted - Skip - No test values generated
10.23.E3 J 0 2 ⁡ ( z ) + 2 ⁢ ∑ k = 1 ∞ J k 2 ⁡ ( z ) = 1 Bessel-J 0 2 𝑧 2 superscript subscript 𝑘 1 Bessel-J 𝑘 2 𝑧 1 {\displaystyle{\displaystyle{J_{0}^{2}}\left(z\right)+2\sum_{k=1}^{\infty}{J_{% k}^{2}}\left(z\right)=1}}
\BesselJ{0}^{2}@{z}+2\sum_{k=1}^{\infty}\BesselJ{k}^{2}@{z} = 1		 ℜ ⁡ ( 0 + k + 1 ) > 0 , ℜ ⁡ ( k + k + 1 ) > 0 formulae-sequence 0 𝑘 1 0 𝑘 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(k+k+1)>0}} 
(BesselJ(0, z))^(2)+ 2*sum((BesselJ(k, z))^(2), k = 1..infinity) = 1

(BesselJ[0, z])^(2)+ 2*Sum[(BesselJ[k, z])^(2), {k, 1, Infinity}, GenerateConditions->None] == 1
 Aborted Successful Successful [Tested: 7] Successful [Tested: 7]
10.23.E4 ∑ k = 0 2 ⁢ n ( - 1 ) k ⁢ J k ⁡ ( z ) ⁢ J 2 ⁢ n - k ⁡ ( z ) + 2 ⁢ ∑ k = 1 ∞ J k ⁡ ( z ) ⁢ J 2 ⁢ n + k ⁡ ( z ) = 0 superscript subscript 𝑘 0 2 𝑛 superscript 1 𝑘 Bessel-J 𝑘 𝑧 Bessel-J 2 𝑛 𝑘 𝑧 2 superscript subscript 𝑘 1 Bessel-J 𝑘 𝑧 Bessel-J 2 𝑛 𝑘 𝑧 0 {\displaystyle{\displaystyle\sum_{k=0}^{2n}(-1)^{k}J_{k}\left(z\right)J_{2n-k}% \left(z\right)\\ +2\sum_{k=1}^{\infty}J_{k}\left(z\right)J_{2n+k}\left(z\right)=0}}
\sum_{k=0}^{2n}(-1)^{k}\BesselJ{k}@{z}\BesselJ{2n-k}@{z}\\ +2\sum_{k=1}^{\infty}\BesselJ{k}@{z}\BesselJ{2n+k}@{z} = 0		 n ≥ 1 , ℜ ⁡ ( k + k + 1 ) > 0 , ℜ ⁡ ( ( 2 ⁢ n - k ) + k + 1 ) > 0 , ℜ ⁡ ( ( 2 ⁢ n + k ) + k + 1 ) > 0 formulae-sequence 𝑛 1 formulae-sequence 𝑘 𝑘 1 0 formulae-sequence 2 𝑛 𝑘 𝑘 1 0 2 𝑛 𝑘 𝑘 1 0 {\displaystyle{\displaystyle n\geq 1,\Re(k+k+1)>0,\Re((2n-k)+k+1)>0,\Re((2n+k)% +k+1)>0}} 
sum((- 1)^(k)* BesselJ(k, z)*BesselJ(2*n - k, z)*; , k = 0..2*n)+ 2*sum(BesselJ(k, z)*BesselJ(2*n + k, z), k = 1..infinity) = 0
 
Sum[(- 1)^(k)* BesselJ[k, z]*BesselJ[2*n - k, z]*, {k, 0, 2*n}, GenerateConditions->None]+ 2*Sum[BesselJ[k, z]*BesselJ[2*n + k, z], {k, 1, Infinity}, GenerateConditions->None] == 0
 Error Failure -
Failed [21 / 21]
Result: Plus[Complex[0.00727987412712798, -0.017853077134921347], Times[2.0, NSum[Times[BesselJ[k, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], BesselJ[Plus[2, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Plus[Complex[2.4034761502300195*^-4, -3.087748713313073*^-5], Times[2.0, NSum[Times[BesselJ[k, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], BesselJ[Plus[4, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.23.E5 ∑ k = 0 n J k ⁡ ( z ) ⁢ J n - k ⁡ ( z ) + 2 ⁢ ∑ k = 1 ∞ ( - 1 ) k ⁢ J k ⁡ ( z ) ⁢ J n + k ⁡ ( z ) = J n ⁡ ( 2 ⁢ z ) superscript subscript 𝑘 0 𝑛 Bessel-J 𝑘 𝑧 Bessel-J 𝑛 𝑘 𝑧 2 superscript subscript 𝑘 1 superscript 1 𝑘 Bessel-J 𝑘 𝑧 Bessel-J 𝑛 𝑘 𝑧 Bessel-J 𝑛 2 𝑧 {\displaystyle{\displaystyle\sum_{k=0}^{n}J_{k}\left(z\right)J_{n-k}\left(z% \right)+2\sum_{k=1}^{\infty}(-1)^{k}J_{k}\left(z\right)J_{n+k}\left(z\right)=J% _{n}\left(2z\right)}}
\sum_{k=0}^{n}\BesselJ{k}@{z}\BesselJ{n-k}@{z}+2\sum_{k=1}^{\infty}(-1)^{k}\BesselJ{k}@{z}\BesselJ{n+k}@{z} = \BesselJ{n}@{2z}		 ℜ ⁡ ( k + k + 1 ) > 0 , ℜ ⁡ ( ( n - k ) + k + 1 ) > 0 , ℜ ⁡ ( ( n + k ) + k + 1 ) > 0 , ℜ ⁡ ( n + k + 1 ) > 0 formulae-sequence 𝑘 𝑘 1 0 formulae-sequence 𝑛 𝑘 𝑘 1 0 formulae-sequence 𝑛 𝑘 𝑘 1 0 𝑛 𝑘 1 0 {\displaystyle{\displaystyle\Re(k+k+1)>0,\Re((n-k)+k+1)>0,\Re((n+k)+k+1)>0,\Re% (n+k+1)>0}} 
sum(BesselJ(k, z)*BesselJ(n - k, z), k = 0..n)+ 2*sum((- 1)^(k)* BesselJ(k, z)*BesselJ(n + k, z), k = 1..infinity) = BesselJ(n, 2*z)
 
Sum[BesselJ[k, z]*BesselJ[n - k, z], {k, 0, n}, GenerateConditions->None]+ 2*Sum[(- 1)^(k)* BesselJ[k, z]*BesselJ[n + k, z], {k, 1, Infinity}, GenerateConditions->None] == BesselJ[n, 2*z]
 Aborted Failure Skipped - Because timed out
Failed [21 / 21]
Result: Plus[Complex[0.024343533040476317, 0.10797471990649704], Times[2.0, NSum[Times[Power[-1, k], BesselJ[k, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], BesselJ[Plus[1, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Plus[Complex[-0.006069425709337772, 0.017711723121060452], Times[2.0, NSum[Times[Power[-1, k], BesselJ[k, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], BesselJ[Plus[2, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.23#Ex1 w = u 2 + v 2 - 2 ⁢ u ⁢ v ⁢ cos ⁡ α 𝑤 superscript 𝑢 2 superscript 𝑣 2 2 𝑢 𝑣 𝛼 {\displaystyle{\displaystyle w=\sqrt{u^{2}+v^{2}-2uv\cos\alpha}}}
w = \sqrt{u^{2}+v^{2}-2uv\cos@@{\alpha}}		 

w = sqrt((u)^(2)+ (v)^(2)- 2*u*v*cos(alpha))
 
w == Sqrt[(u)^(2)+ (v)^(2)- 2*u*v*Cos[\[Alpha]]]
 Failure Failure
Failed [300 / 300]
Result: -.3146075610-.1816387601*I
Test Values: {alpha = 3/2, u = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I}

Result: -1.680632965+.1843866439*I
Test Values: {alpha = 3/2, u = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, w = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.3146075609842255, -0.18163876002333418]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5]}

Result: Complex[0.4375091763619045, 0.252596040745477]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 0.5]}

... skip entries to safe data
10.23#Ex2 u - v ⁢ cos ⁡ α = w ⁢ cos ⁡ χ 𝑢 𝑣 𝛼 𝑤 𝜒 {\displaystyle{\displaystyle u-v\cos\alpha=w\cos\chi}}
u-v\cos@@{\alpha} = w\cos@@{\chi}		 

u - v*cos(alpha) = w*cos(chi)
 
u - v*Cos[\[Alpha]] == w*Cos[\[Chi]]
 Failure Failure
Failed [300 / 300]
Result: -.263783978e-1+.4431282844*I
Test Values: {alpha = 3/2, chi = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I}

Result: .8262683052-.3665121890*I
Test Values: {alpha = 3/2, chi = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, w = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.026378398027867456, 0.44312828415668515]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-0.023973249213014358, -0.5554825514041751]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.23#Ex3 v ⁢ sin ⁡ α = w ⁢ sin ⁡ χ 𝑣 𝛼 𝑤 𝜒 {\displaystyle{\displaystyle v\sin\alpha=w\sin\chi}}
v\sin@@{\alpha} = w\sin@@{\chi}		 

v*sin(alpha) = w*sin(chi)
 
v*Sin[\[Alpha]] == w*Sin[\[Chi]]
 Failure Failure
Failed [300 / 300]
Result: .2887554391-.2231097873*I
Test Values: {alpha = 3/2, chi = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I}

Result: 1.585713279-.763530664e-1*I
Test Values: {alpha = 3/2, chi = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, w = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [294 / 300]
Result: Complex[0.2887554393029954, -0.22310978722682606]
Test Values: {Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[1.8740447527972026, 0.09051196331992012]
Test Values: {Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.23.E9 e i ⁢ v ⁢ cos ⁡ α = Γ ⁡ ( ν ) ( 1 2 ⁢ v ) ν ⁢ ∑ k = 0 ∞ ( ν + k ) ⁢ i k ⁢ J ν + k ⁡ ( v ) ⁢ C k ( ν ) ⁡ ( cos ⁡ α ) superscript 𝑒 𝑖 𝑣 𝛼 Euler-Gamma 𝜈 superscript 1 2 𝑣 𝜈 superscript subscript 𝑘 0 𝜈 𝑘 superscript 𝑖 𝑘 Bessel-J 𝜈 𝑘 𝑣 ultraspherical-Gegenbauer-polynomial 𝜈 𝑘 𝛼 {\displaystyle{\displaystyle e^{iv\cos\alpha}=\frac{\Gamma\left(\nu\right)}{(% \tfrac{1}{2}v)^{\nu}}\*\sum_{k=0}^{\infty}(\nu+k)i^{k}J_{\nu+k}\left(v\right)C% ^{(\nu)}_{k}\left(\cos\alpha\right)}}
e^{iv\cos@@{\alpha}} = \frac{\EulerGamma@{\nu}}{(\tfrac{1}{2}v)^{\nu}}\*\sum_{k=0}^{\infty}(\nu+k)i^{k}\BesselJ{\nu+k}@{v}\ultrasphpoly{\nu}{k}@{\cos@@{\alpha}}		 ℜ ⁡ ( ( ν + k ) + k + 1 ) > 0 , ℜ ⁡ ( ν ) > 0 formulae-sequence 𝜈 𝑘 𝑘 1 0 𝜈 0 {\displaystyle{\displaystyle\Re((\nu+k)+k+1)>0,\Re(\nu)>0}} 
exp(I*v*cos(alpha)) = (GAMMA(nu))/(((1)/(2)*v)^(nu))* sum((nu + k)*(I)^(k)* BesselJ(nu + k, v)*GegenbauerC(k, nu, cos(alpha)), k = 0..infinity)
 
Exp[I*v*Cos[\[Alpha]]] == Divide[Gamma[\[Nu]],(Divide[1,2]*v)^\[Nu]]* Sum[(\[Nu]+ k)*(I)^(k)* BesselJ[\[Nu]+ k, v]*GegenbauerC[k, \[Nu], Cos[\[Alpha]]], {k, 0, Infinity}, GenerateConditions->None]
 Aborted Failure Skipped - Because timed out Skipped - Because timed out
10.23.E15 ( 1 2 ⁢ z ) ν = ∑ k = 0 ∞ ( ν + 2 ⁢ k ) ⁢ Γ ⁡ ( ν + k ) k ! ⁢ J ν + 2 ⁢ k ⁡ ( z ) superscript 1 2 𝑧 𝜈 superscript subscript 𝑘 0 𝜈 2 𝑘 Euler-Gamma 𝜈 𝑘 𝑘 Bessel-J 𝜈 2 𝑘 𝑧 {\displaystyle{\displaystyle(\tfrac{1}{2}z)^{\nu}=\sum_{k=0}^{\infty}\frac{(% \nu+2k)\Gamma\left(\nu+k\right)}{k!}J_{\nu+2k}\left(z\right)}}
(\tfrac{1}{2}z)^{\nu} = \sum_{k=0}^{\infty}\frac{(\nu+2k)\EulerGamma@{\nu+k}}{k!}\BesselJ{\nu+2k}@{z}		 ℜ ⁡ ( ( ν + 2 ⁢ k ) + k + 1 ) > 0 , ℜ ⁡ ( ν + k ) > 0 formulae-sequence 𝜈 2 𝑘 𝑘 1 0 𝜈 𝑘 0 {\displaystyle{\displaystyle\Re((\nu+2k)+k+1)>0,\Re(\nu+k)>0}} 
((1)/(2)*z)^(nu) = sum(((nu + 2*k)*GAMMA(nu + k))/(factorial(k))*BesselJ(nu + 2*k, z), k = 0..infinity)
 
(Divide[1,2]*z)^\[Nu] == Sum[Divide[(\[Nu]+ 2*k)*Gamma[\[Nu]+ k],(k)!]*BesselJ[\[Nu]+ 2*k, z], {k, 0, Infinity}, GenerateConditions->None]
 Aborted Successful Skipped - Because timed out Successful [Tested: 7]
10.23.E16 Y 0 ⁡ ( z ) = 2 π ⁢ ( ln ⁡ ( 1 2 ⁢ z ) + γ ) ⁢ J 0 ⁡ ( z ) - 4 π ⁢ ∑ k = 1 ∞ ( - 1 ) k ⁢ J 2 ⁢ k ⁡ ( z ) k Bessel-Y-Weber 0 𝑧 2 𝜋 1 2 𝑧 Bessel-J 0 𝑧 4 𝜋 superscript subscript 𝑘 1 superscript 1 𝑘 Bessel-J 2 𝑘 𝑧 𝑘 {\displaystyle{\displaystyle Y_{0}\left(z\right)=\frac{2}{\pi}\left(\ln\left(% \tfrac{1}{2}z\right)+\gamma\right)J_{0}\left(z\right)-\frac{4}{\pi}\sum_{k=1}^% {\infty}(-1)^{k}\frac{J_{2k}\left(z\right)}{k}}}
\BesselY{0}@{z} = \frac{2}{\pi}\left(\ln@{\tfrac{1}{2}z}+\EulerConstant\right)\BesselJ{0}@{z}-\frac{4}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{\BesselJ{2k}@{z}}{k}		 ℜ ⁡ ( 0 + k + 1 ) > 0 , ℜ ⁡ ( ( 2 ⁢ k ) + k + 1 ) > 0 , ℜ ⁡ ( ( - 0 ) + k + 1 ) > 0 formulae-sequence 0 𝑘 1 0 formulae-sequence 2 𝑘 𝑘 1 0 0 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0,\Re((2k)+k+1)>0,\Re((-0)+k+1)>0}} 
BesselY(0, z) = (2)/(Pi)*(ln((1)/(2)*z)+ gamma)*BesselJ(0, z)-(4)/(Pi)*sum((- 1)^(k)*(BesselJ(2*k, z))/(k), k = 1..infinity)
 
BesselY[0, z] == Divide[2,Pi]*(Log[Divide[1,2]*z]+ EulerGamma)*BesselJ[0, z]-Divide[4,Pi]*Sum[(- 1)^(k)*Divide[BesselJ[2*k, z],k], {k, 1, Infinity}, GenerateConditions->None]
 Aborted Successful Successful [Tested: 7] Successful [Tested: 7]
10.23.E17 Y n ⁡ ( z ) = - n ! ⁢ ( 1 2 ⁢ z ) - n π ⁢ ∑ k = 0 n - 1 ( 1 2 ⁢ z ) k ⁢ J k ⁡ ( z ) k ! ⁢ ( n - k ) + 2 π ⁢ ( ln ⁡ ( 1 2 ⁢ z ) - ψ ⁡ ( n + 1 ) ) ⁢ J n ⁡ ( z ) - 2 π ⁢ ∑ k = 1 ∞ ( - 1 ) k ⁢ ( n + 2 ⁢ k ) ⁢ J n + 2 ⁢ k ⁡ ( z ) k ⁢ ( n + k ) Bessel-Y-Weber 𝑛 𝑧 𝑛 superscript 1 2 𝑧 𝑛 𝜋 superscript subscript 𝑘 0 𝑛 1 superscript 1 2 𝑧 𝑘 Bessel-J 𝑘 𝑧 𝑘 𝑛 𝑘 2 𝜋 1 2 𝑧 digamma 𝑛 1 Bessel-J 𝑛 𝑧 2 𝜋 superscript subscript 𝑘 1 superscript 1 𝑘 𝑛 2 𝑘 Bessel-J 𝑛 2 𝑘 𝑧 𝑘 𝑛 𝑘 {\displaystyle{\displaystyle Y_{n}\left(z\right)=-\frac{n!(\tfrac{1}{2}z)^{-n}% }{\pi}\sum_{k=0}^{n-1}\frac{(\tfrac{1}{2}z)^{k}J_{k}\left(z\right)}{k!(n-k)}+% \frac{2}{\pi}\left(\ln\left(\tfrac{1}{2}z\right)-\psi\left(n+1\right)\right)J_% {n}\left(z\right)-\frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{(n+2k)J_{n+2k}% \left(z\right)}{k(n+k)}}}
\BesselY{n}@{z} = -\frac{n!(\tfrac{1}{2}z)^{-n}}{\pi}\sum_{k=0}^{n-1}\frac{(\tfrac{1}{2}z)^{k}\BesselJ{k}@{z}}{k!(n-k)}+\frac{2}{\pi}\left(\ln@{\tfrac{1}{2}z}-\digamma@{n+1}\right)\BesselJ{n}@{z}-\frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{(n+2k)\BesselJ{n+2k}@{z}}{k(n+k)}		 ℜ ⁡ ( n + k + 1 ) > 0 , ℜ ⁡ ( k + k + 1 ) > 0 , ℜ ⁡ ( ( n + 2 ⁢ k ) + k + 1 ) > 0 , ℜ ⁡ ( ( - n ) + k + 1 ) > 0 formulae-sequence 𝑛 𝑘 1 0 formulae-sequence 𝑘 𝑘 1 0 formulae-sequence 𝑛 2 𝑘 𝑘 1 0 𝑛 𝑘 1 0 {\displaystyle{\displaystyle\Re(n+k+1)>0,\Re(k+k+1)>0,\Re((n+2k)+k+1)>0,\Re((-% n)+k+1)>0}} 
BesselY(n, z) = -(factorial(n)*((1)/(2)*z)^(- n))/(Pi)*sum((((1)/(2)*z)^(k)* BesselJ(k, z))/(factorial(k)*(n - k)), k = 0..n - 1)+(2)/(Pi)*(ln((1)/(2)*z)- Psi(n + 1))*BesselJ(n, z)-(2)/(Pi)*sum((- 1)^(k)*((n + 2*k)*BesselJ(n + 2*k, z))/(k*(n + k)), k = 1..infinity)
 
BesselY[n, z] == -Divide[(n)!*(Divide[1,2]*z)^(- n),Pi]*Sum[Divide[(Divide[1,2]*z)^(k)* BesselJ[k, z],(k)!*(n - k)], {k, 0, n - 1}, GenerateConditions->None]+Divide[2,Pi]*(Log[Divide[1,2]*z]- PolyGamma[n + 1])*BesselJ[n, z]-Divide[2,Pi]*Sum[(- 1)^(k)*Divide[(n + 2*k)*BesselJ[n + 2*k, z],k*(n + k)], {k, 1, Infinity}, GenerateConditions->None]
 Aborted Failure Manual Skip!
Failed [16 / 21]
Result: Plus[Complex[-0.41373222494160333, 0.38808044477324316], Times[Complex[0.5513288954217921, -0.31830988618379064], DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[Times[-1, ], 1], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], []], Times[Plus[4, Times[12, ], Times[12, Power[, 2]], Times[4, Power[, 3]], Times[-4, 1], Times[-8, , 1], Times[-4, Power[, 2], 1], Times[, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-1, 1, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], [Plus[1, ]]], Times[4, Plus[1, ], Plus[-5, Times[-6, ], Times[-2, Power[, 2]], Times[3, 1], Times[2, , 1]], [Plus[2, ]]], Times[-4, Plus[1, ], Plus[2, ], Plus[-2, Times[-1, ], 1], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Power[1, -1], BesselJ[0, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[2], Plus[Times[Power[1, -1], BesselJ[0, Power[E, Times[Complex[0, Rational[1, 6]], Pi]<syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.6198631863998064, 5.383408526303685], Times[Complex[0.0, -15.278874536821952], DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-1, Power[-1, Rational[1, 3]], Plus[-3, ], []], Times[Plus[-8, Times[-3, Power[-1, Rational[1, 3]]], Times[-12, ], Times[Power[-1, Rational[1, 3]], ], Times[4, Power[, 3]]], [Plus[1, ]]], Times[-8, Plus[1, ], Plus[-2, Power[, 2]], [Plus[2, ]]], Times[4, Plus[-1, ], Plus[1, ], Plus[2, ], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Rational[1, 3], BesselJ[0, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[2], Plus[Times[Rational[1, 3], BesselJ[0, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[Rational[1, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], BesselJ[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]}]][3.0]]], {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.24.E1 x 2 ⁢ d 2 w d x 2 + x ⁢ d w d x + ( x 2 + ν 2 ) ⁢ w = 0 superscript 𝑥 2 derivative 𝑤 𝑥 2 𝑥 derivative 𝑤 𝑥 superscript 𝑥 2 superscript 𝜈 2 𝑤 0 {\displaystyle{\displaystyle x^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}+% x\frac{\mathrm{d}w}{\mathrm{d}x}+(x^{2}+\nu^{2})w=0}}
x^{2}\deriv[2]{w}{x}+x\deriv{w}{x}+(x^{2}+\nu^{2})w = 0		 

(x)^(2)* diff(w, [x$(2)])+ x*diff(w, x)+((x)^(2)+ (nu)^(2))*w = 0
 
(x)^(2)* D[w, {x, 2}]+ x*D[w, x]+((x)^(2)+ \[Nu]^(2))*w == 0
 Failure Failure
Failed [300 / 300]
Result: 1.948557159+2.125000000*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 3/2}

Result: .2165063513+1.125000001*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.9485571585149875, 2.125]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[1.948557158514987, 0.12499999999999989]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.24#Ex1 J ~ ν ⁡ ( x ) = sech ⁡ ( 1 2 ⁢ π ⁢ ν ) ⁢ ℜ ⁡ ( J i ⁢ ν ⁡ ( x ) ) Bessel-J-imaginary-order 𝜈 𝑥 1 2 𝜋 𝜈 Bessel-J 𝑖 𝜈 𝑥 {\displaystyle{\displaystyle\widetilde{J}_{\nu}\left(x\right)=\operatorname{% sech}\left(\tfrac{1}{2}\pi\nu\right)\Re\left(J_{i\nu}\left(x\right)\right)}}
\BesselJimag{\nu}@{x} = \sech@{\tfrac{1}{2}\pi\nu}\realpart@{\BesselJ{i\nu}@{x}}		 ℜ ⁡ ( ( i ⁢ ν ) + k + 1 ) > 0 imaginary-unit 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re((\mathrm{i}\nu)+k+1)>0}} 
sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x)) = sech((1)/(2)*Pi*nu)*Re(BesselJ(I*nu, x))
 
Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]] == Sech[Divide[1,2]*Pi*\[Nu]]*Re[BesselJ[I*\[Nu], x]]
 Successful Successful - Successful [Tested: 30]
10.24#Ex2 Y ~ ν ⁡ ( x ) = sech ⁡ ( 1 2 ⁢ π ⁢ ν ) ⁢ ℜ ⁡ ( Y i ⁢ ν ⁡ ( x ) ) Bessel-Y-Weber-imaginary-order 𝜈 𝑥 1 2 𝜋 𝜈 Bessel-Y-Weber 𝑖 𝜈 𝑥 {\displaystyle{\displaystyle\widetilde{Y}_{\nu}\left(x\right)=\operatorname{% sech}\left(\tfrac{1}{2}\pi\nu\right)\Re\left(Y_{i\nu}\left(x\right)\right)}}
\BesselYimag{\nu}@{x} = \sech@{\tfrac{1}{2}\pi\nu}\realpart@{\BesselY{i\nu}@{x}}		 ℜ ⁡ ( ( i ⁢ ν ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( i ⁢ ν ) ) + k + 1 ) > 0 formulae-sequence imaginary-unit 𝜈 𝑘 1 0 imaginary-unit 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re((\mathrm{i}\nu)+k+1)>0,\Re((-(\mathrm{i}\nu))+% k+1)>0}} 
sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x)) = sech((1)/(2)*Pi*nu)*Re(BesselY(I*nu, x))
 
Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]] == Sech[Divide[1,2]*Pi*\[Nu]]*Re[BesselY[I*\[Nu], x]]
 Successful Successful - Successful [Tested: 30]
10.24.E3 Γ ⁡ ( 1 + i ⁢ ν ) = ( π ⁢ ν sinh ⁡ ( π ⁢ ν ) ) 1 2 ⁢ e i ⁢ γ ν Euler-Gamma 1 𝑖 𝜈 superscript 𝜋 𝜈 𝜋 𝜈 1 2 superscript 𝑒 𝑖 subscript 𝛾 𝜈 {\displaystyle{\displaystyle\Gamma\left(1+i\nu\right)=\left(\frac{\pi\nu}{% \sinh\left(\pi\nu\right)}\right)^{\frac{1}{2}}e^{i\gamma_{\nu}}}}
\EulerGamma@{1+i\nu} = \left(\frac{\pi\nu}{\sinh@{\pi\nu}}\right)^{\frac{1}{2}}e^{i\gamma_{\nu}}		 ℜ ⁡ ( 1 + i ⁢ ν ) > 0 1 imaginary-unit 𝜈 0 {\displaystyle{\displaystyle\Re(1+\mathrm{i}\nu)>0}} 
GAMMA(1 + I*nu) = ((Pi*nu)/(sinh(Pi*nu)))^((1)/(2))* exp(I*gamma[nu])
 
Gamma[1 + I*\[Nu]] == (Divide[Pi*\[Nu],Sinh[Pi*\[Nu]]])^(Divide[1,2])* Exp[I*Subscript[\[Gamma], \[Nu]]]
 Failure Failure
Failed [300 / 300]
Result: .131682196e-1-.6479738907*I
Test Values: {gamma = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, gamma[nu] = 1/2*3^(1/2)+1/2*I}

Result: .2393622021-.2867640040*I
Test Values: {gamma = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, gamma[nu] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.013168219691258531, -0.6479738909120968]
Test Values: {Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[γ, ν], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[0.23936220222535412, -0.28676400411697583]
Test Values: {Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[γ, ν], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.24#Ex3 J ~ - ν ⁡ ( x ) = J ~ ν ⁡ ( x ) Bessel-J-imaginary-order 𝜈 𝑥 Bessel-J-imaginary-order 𝜈 𝑥 {\displaystyle{\displaystyle\widetilde{J}_{-\nu}\left(x\right)=\widetilde{J}_{% \nu}\left(x\right)}}
\BesselJimag{-\nu}@{x} = \BesselJimag{\nu}@{x}		 

sech((1/2)*Pi*(- nu))*Re(BesselJ(I*(- nu), x)) = sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x))
 
Sech[1/2 Pi - \[Nu]] Re[BesselJ[I - \[Nu], x]] == Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]]
 Failure Failure
Failed [12 / 30]
Result: .1765981285-.1547836875*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2}

Result: -1.059084556+.9282601935*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[-0.6353785354467336, 0.04153700144653363]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[0.2910880978413849, 0.681683596996288]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.24#Ex4 Y ~ - ν ⁡ ( x ) = Y ~ ν ⁡ ( x ) Bessel-Y-Weber-imaginary-order 𝜈 𝑥 Bessel-Y-Weber-imaginary-order 𝜈 𝑥 {\displaystyle{\displaystyle\widetilde{Y}_{-\nu}\left(x\right)=\widetilde{Y}_{% \nu}\left(x\right)}}
\BesselYimag{-\nu}@{x} = \BesselYimag{\nu}@{x}		 ℜ ⁡ ( ( i ⁢ ( - ν ) ) + k + 1 ) > 0 , ℜ ⁡ ( ( i ⁢ ν ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( i ⁢ ( - ν ) ) ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( i ⁢ ν ) ) + k + 1 ) > 0 formulae-sequence imaginary-unit 𝜈 𝑘 1 0 formulae-sequence imaginary-unit 𝜈 𝑘 1 0 formulae-sequence imaginary-unit 𝜈 𝑘 1 0 imaginary-unit 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re((\mathrm{i}(-\nu))+k+1)>0,\Re((\mathrm{i}\nu)+% k+1)>0,\Re((-(\mathrm{i}(-\nu)))+k+1)>0,\Re((-(\mathrm{i}\nu))+k+1)>0}} 
sech((1/2)*Pi*(- nu))*Re(BesselY(I*(- nu), x)) = sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x))
 
Sech[1/2 Pi - \[Nu]] Re[BesselY[I - \[Nu], x]] == Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]]
 Failure Failure
Failed [12 / 30]
Result: -.6730010946+.5898680353*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2}

Result: -.1980888923+.1736197856*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[0.16541121369118172, 0.7534126929509344]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-0.3242468905843751, -0.9796849117084342]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.24.E5 𝒲 ⁡ { J ~ ν ⁡ ( x ) , Y ~ ν ⁡ ( x ) } = 2 / ( π ⁢ x ) Wronskian Bessel-J-imaginary-order 𝜈 𝑥 Bessel-Y-Weber-imaginary-order 𝜈 𝑥 2 𝜋 𝑥 {\displaystyle{\displaystyle\mathscr{W}\left\{\widetilde{J}_{\nu}\left(x\right% ),\widetilde{Y}_{\nu}\left(x\right)\right\}=2/(\pi x)}}
\Wronskian@{\BesselJimag{\nu}@{x},\BesselYimag{\nu}@{x}} = 2/(\pi x)		 ℜ ⁡ ( ( i ⁢ ν ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( i ⁢ ν ) ) + k + 1 ) > 0 formulae-sequence imaginary-unit 𝜈 𝑘 1 0 imaginary-unit 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re((\mathrm{i}\nu)+k+1)>0,\Re((-(\mathrm{i}\nu))+% k+1)>0}} 
(sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x)))*diff(sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x)), x)-diff(sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x)), x)*(sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x))) = 2/(Pi*x)
 
Wronskian[{Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]], Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]]}, x] == 2/(Pi*x)
 Failure Failure
Failed [12 / 30]
Result: -.3214564733-.7786157192*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2}

Result: -.6431025084-4.765445687*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [30 / 30]
Result: Plus[-0.4244131815783876, Times[Complex[0.017184424665049866, -0.12995814793225188], Plus[Times[Complex[5.94457417937745, -0.08806734388290616], Derivative[1][Re][Complex[0.5424102683642863, 1.3820413572565333]]], Times[Complex[0.04670634387761448, 2.0064149502593187], Derivative[1][Re][Complex[1.5013396639532606, -0.5145465005058608]]]]]]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Plus[-0.4244131815783876, Times[Complex[-0.5062208144169521, 0.3689208146583662], Plus[Times[Complex[1.2690034139339206, -1.428145592425075], Derivative[1][Re][Complex[-0.5230512553281585, -0.7250724679588263]]], Times[Complex[0.9907135967899046, 0.5862869255257461], Derivative[1][Re][Complex[0.9118063408652576, -0.381897212811936]]]]]]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.24.E9 Y ~ 0 ⁡ ( x ) = Y 0 ⁡ ( x ) Bessel-Y-Weber-imaginary-order 0 𝑥 Bessel-Y-Weber 0 𝑥 {\displaystyle{\displaystyle\widetilde{Y}_{0}\left(x\right)=Y_{0}\left(x\right% )}}
\BesselYimag{0}@{x} = \BesselY{0}@{x}		 ℜ ⁡ ( 0 + k + 1 ) > 0 , ℜ ⁡ ( ( - 0 ) + k + 1 ) > 0 , ℜ ⁡ ( ( i ⁢ 0 ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( i ⁢ 0 ) ) + k + 1 ) > 0 formulae-sequence 0 𝑘 1 0 formulae-sequence 0 𝑘 1 0 formulae-sequence imaginary-unit 0 𝑘 1 0 imaginary-unit 0 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0,\Re((-0)+k+1)>0,\Re((\mathrm{i}0)+k+1% )>0,\Re((-(\mathrm{i}0))+k+1)>0}} 
sech((1/2)*Pi*(0))*Re(BesselY(I*(0), x)) = BesselY(0, x)
 
Sech[1/2 Pi 0] Re[BesselY[I 0, x]] == BesselY[0, x]
 Failure Failure Successful [Tested: 3] Successful [Tested: 3]
10.25.E1 z 2 ⁢ d 2 w d z 2 + z ⁢ d w d z - ( z 2 + ν 2 ) ⁢ w = 0 superscript 𝑧 2 derivative 𝑤 𝑧 2 𝑧 derivative 𝑤 𝑧 superscript 𝑧 2 superscript 𝜈 2 𝑤 0 {\displaystyle{\displaystyle z^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+% z\frac{\mathrm{d}w}{\mathrm{d}z}-(z^{2}+\nu^{2})w=0}}
z^{2}\deriv[2]{w}{z}+z\deriv{w}{z}-(z^{2}+\nu^{2})w = 0		 

(z)^(2)* diff(w, [z$(2)])+ z*diff(w, z)-((z)^(2)+ (nu)^(2))*w = 0
 
(z)^(2)* D[w, {z, 2}]+ z*D[w, z]-((z)^(2)+ \[Nu]^(2))*w == 0
 Failure Failure
Failed [220 / 300]
Result: -.6467477718e-9-2.000000002*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}

Result: -.8660254040e-9-2.000000001*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [264 / 300]
Result: Complex[0.0, -2.0]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[0.0, -2.0]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
10.25.E2 I ν ⁡ ( z ) = ( 1 2 ⁢ z ) ν ⁢ ∑ k = 0 ∞ ( 1 4 ⁢ z 2 ) k k ! ⁢ Γ ⁡ ( ν + k + 1 ) modified-Bessel-first-kind 𝜈 𝑧 superscript 1 2 𝑧 𝜈 superscript subscript 𝑘 0 superscript 1 4 superscript 𝑧 2 𝑘 𝑘 Euler-Gamma 𝜈 𝑘 1 {\displaystyle{\displaystyle I_{\nu}\left(z\right)=(\tfrac{1}{2}z)^{\nu}\sum_{% k=0}^{\infty}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!\Gamma\left(\nu+k+1\right)}}}
\modBesselI{\nu}@{z} = (\tfrac{1}{2}z)^{\nu}\sum_{k=0}^{\infty}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!\EulerGamma@{\nu+k+1}}		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
BesselI(nu, z) = ((1)/(2)*z)^(nu)* sum((((1)/(4)*(z)^(2))^(k))/(factorial(k)*GAMMA(nu + k + 1)), k = 0..infinity)
 
BesselI[\[Nu], z] == (Divide[1,2]*z)^\[Nu]* Sum[Divide[(Divide[1,4]*(z)^(2))^(k),(k)!*Gamma[\[Nu]+ k + 1]], {k, 0, Infinity}, GenerateConditions->None]
 Successful Successful - Successful [Tested: 70]
10.27.E1 I - n ⁡ ( z ) = I n ⁡ ( z ) modified-Bessel-first-kind 𝑛 𝑧 modified-Bessel-first-kind 𝑛 𝑧 {\displaystyle{\displaystyle I_{-n}\left(z\right)=I_{n}\left(z\right)}}
\modBesselI{-n}@{z} = \modBesselI{n}@{z}		 ℜ ⁡ ( ( - n ) + k + 1 ) > 0 , ℜ ⁡ ( n + k + 1 ) > 0 formulae-sequence 𝑛 𝑘 1 0 𝑛 𝑘 1 0 {\displaystyle{\displaystyle\Re((-n)+k+1)>0,\Re(n+k+1)>0}} 
BesselI(- n, z) = BesselI(n, z)
 
BesselI[- n, z] == BesselI[n, z]
 Failure Failure Successful [Tested: 21] Successful [Tested: 21]
10.27.E2 I - ν ⁡ ( z ) = I ν ⁡ ( z ) + ( 2 / π ) ⁢ sin ⁡ ( ν ⁢ π ) ⁢ K ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 modified-Bessel-first-kind 𝜈 𝑧 2 𝜋 𝜈 𝜋 modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{-\nu}\left(z\right)=I_{\nu}\left(z\right)+(2/% \pi)\sin\left(\nu\pi\right)K_{\nu}\left(z\right)}}
\modBesselI{-\nu}@{z} = \modBesselI{\nu}@{z}+(2/\pi)\sin@{\nu\pi}\modBesselK{\nu}@{z}		 ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re((-\nu)+k+1)>0,\Re(\nu+k+1)>0}} 
BesselI(- nu, z) = BesselI(nu, z)+(2/Pi)*sin(nu*Pi)*BesselK(nu, z)
 
BesselI[- \[Nu], z] == BesselI[\[Nu], z]+(2/Pi)*Sin[\[Nu]*Pi]*BesselK[\[Nu], z]
 Successful Successful - Successful [Tested: 70]
10.27.E3 K - ν ⁡ ( z ) = K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{-\nu}\left(z\right)=K_{\nu}\left(z\right)}}
\modBesselK{-\nu}@{z} = \modBesselK{\nu}@{z}		 

BesselK(- nu, z) = BesselK(nu, z)
 
BesselK[- \[Nu], z] == BesselK[\[Nu], z]
 Successful Successful - Successful [Tested: 70]
10.27.E4 K ν ⁡ ( z ) = 1 2 ⁢ π ⁢ I - ν ⁡ ( z ) - I ν ⁡ ( z ) sin ⁡ ( ν ⁢ π ) modified-Bessel-second-kind 𝜈 𝑧 1 2 𝜋 modified-Bessel-first-kind 𝜈 𝑧 modified-Bessel-first-kind 𝜈 𝑧 𝜈 𝜋 {\displaystyle{\displaystyle K_{\nu}\left(z\right)=\tfrac{1}{2}\pi\frac{I_{-% \nu}\left(z\right)-I_{\nu}\left(z\right)}{\sin\left(\nu\pi\right)}}}
\modBesselK{\nu}@{z} = \tfrac{1}{2}\pi\frac{\modBesselI{-\nu}@{z}-\modBesselI{\nu}@{z}}{\sin@{\nu\pi}}		 ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re((-\nu)+k+1)>0,\Re(\nu+k+1)>0}} 
BesselK(nu, z) = (1)/(2)*Pi*(BesselI(- nu, z)- BesselI(nu, z))/(sin(nu*Pi))
 
BesselK[\[Nu], z] == Divide[1,2]*Pi*Divide[BesselI[- \[Nu], z]- BesselI[\[Nu], z],Sin[\[Nu]*Pi]]
 Successful Successful -
Failed [14 / 70]
Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]}

Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]}

... skip entries to safe data
10.27.E6 I ν ⁡ ( z ) = e - ν ⁢ π ⁢ i / 2 ⁢ J ν ⁡ ( z ⁢ e + π ⁢ i / 2 ) modified-Bessel-first-kind 𝜈 𝑧 superscript 𝑒 𝜈 𝜋 𝑖 2 Bessel-J 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 2 {\displaystyle{\displaystyle I_{\nu}\left(z\right)=e^{-\nu\pi i/2}J_{\nu}\left% (ze^{+\pi i/2}\right)}}
\modBesselI{\nu}@{z} = e^{-\nu\pi i/2}\BesselJ{\nu}@{ze^{+\pi i/2}}		 - π ≤ + ph ⁡ z , - π ≤ - ph ⁡ z , + ph ⁡ z ≤ 1 2 ⁢ π , - ph ⁡ z ≤ 1 2 ⁢ π , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜋 phase 𝑧 formulae-sequence 𝜋 phase 𝑧 formulae-sequence phase 𝑧 1 2 𝜋 formulae-sequence phase 𝑧 1 2 𝜋 𝜈 𝑘 1 0 {\displaystyle{\displaystyle-\pi\leq+\operatorname{ph}z,-\pi\leq-\operatorname% {ph}z,+\operatorname{ph}z\leq\tfrac{1}{2}\pi,-\operatorname{ph}z\leq\tfrac{1}{% 2}\pi,\Re(\nu+k+1)>0}} 
BesselI(nu, z) = exp(- nu*Pi*I/2)*BesselJ(nu, z*exp(+ Pi*I/2))
 
BesselI[\[Nu], z] == Exp[- \[Nu]*Pi*I/2]*BesselJ[\[Nu], z*Exp[+ Pi*I/2]]
 Failure Failure Successful [Tested: 50] Successful [Tested: 50]
10.27.E6 I ν ⁡ ( z ) = e + ν ⁢ π ⁢ i / 2 ⁢ J ν ⁡ ( z ⁢ e - π ⁢ i / 2 ) modified-Bessel-first-kind 𝜈 𝑧 superscript 𝑒 𝜈 𝜋 𝑖 2 Bessel-J 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 2 {\displaystyle{\displaystyle I_{\nu}\left(z\right)=e^{+\nu\pi i/2}J_{\nu}\left% (ze^{-\pi i/2}\right)}}
\modBesselI{\nu}@{z} = e^{+\nu\pi i/2}\BesselJ{\nu}@{ze^{-\pi i/2}}		 - π ≤ + ph ⁡ z , - π ≤ - ph ⁡ z , + ph ⁡ z ≤ 1 2 ⁢ π , - ph ⁡ z ≤ 1 2 ⁢ π , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜋 phase 𝑧 formulae-sequence 𝜋 phase 𝑧 formulae-sequence phase 𝑧 1 2 𝜋 formulae-sequence phase 𝑧 1 2 𝜋 𝜈 𝑘 1 0 {\displaystyle{\displaystyle-\pi\leq+\operatorname{ph}z,-\pi\leq-\operatorname% {ph}z,+\operatorname{ph}z\leq\tfrac{1}{2}\pi,-\operatorname{ph}z\leq\tfrac{1}{% 2}\pi,\Re(\nu+k+1)>0}} 
BesselI(nu, z) = exp(+ nu*Pi*I/2)*BesselJ(nu, z*exp(- Pi*I/2))
 
BesselI[\[Nu], z] == Exp[+ \[Nu]*Pi*I/2]*BesselJ[\[Nu], z*Exp[- Pi*I/2]]
 Failure Failure Successful [Tested: 50] Successful [Tested: 50]
10.27.E7 I ν ⁡ ( z ) = 1 2 ⁢ e - ν ⁢ π ⁢ i / 2 ⁢ ( H ν ( 1 ) ⁡ ( z ⁢ e + π ⁢ i / 2 ) + H ν ( 2 ) ⁡ ( z ⁢ e + π ⁢ i / 2 ) ) modified-Bessel-first-kind 𝜈 𝑧 1 2 superscript 𝑒 𝜈 𝜋 𝑖 2 Hankel-H-1-Bessel-third-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 2 Hankel-H-2-Bessel-third-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 2 {\displaystyle{\displaystyle I_{\nu}\left(z\right)=\tfrac{1}{2}e^{-\nu\pi i/2}% \left({H^{(1)}_{\nu}}\left(ze^{+\pi i/2}\right)+{H^{(2)}_{\nu}}\left(ze^{+\pi i% /2}\right)\right)}}
\modBesselI{\nu}@{z} = \tfrac{1}{2}e^{-\nu\pi i/2}\left(\HankelH{1}{\nu}@{ze^{+\pi i/2}}+\HankelH{2}{\nu}@{ze^{+\pi i/2}}\right)		 - π ≤ + ph ⁡ z , - π ≤ - ph ⁡ z , + ph ⁡ z ≤ 1 2 ⁢ π , - ph ⁡ z ≤ 1 2 ⁢ π , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜋 phase 𝑧 formulae-sequence 𝜋 phase 𝑧 formulae-sequence phase 𝑧 1 2 𝜋 formulae-sequence phase 𝑧 1 2 𝜋 𝜈 𝑘 1 0 {\displaystyle{\displaystyle-\pi\leq+\operatorname{ph}z,-\pi\leq-\operatorname% {ph}z,+\operatorname{ph}z\leq\tfrac{1}{2}\pi,-\operatorname{ph}z\leq\tfrac{1}{% 2}\pi,\Re(\nu+k+1)>0}} 
BesselI(nu, z) = (1)/(2)*exp(- nu*Pi*I/2)*(HankelH1(nu, z*exp(+ Pi*I/2))+ HankelH2(nu, z*exp(+ Pi*I/2)))
 
BesselI[\[Nu], z] == Divide[1,2]*Exp[- \[Nu]*Pi*I/2]*(HankelH1[\[Nu], z*Exp[+ Pi*I/2]]+ HankelH2[\[Nu], z*Exp[+ Pi*I/2]])
 Failure Failure Successful [Tested: 50] Successful [Tested: 50]
10.27.E7 I ν ⁡ ( z ) = 1 2 ⁢ e + ν ⁢ π ⁢ i / 2 ⁢ ( H ν ( 1 ) ⁡ ( z ⁢ e - π ⁢ i / 2 ) + H ν ( 2 ) ⁡ ( z ⁢ e - π ⁢ i / 2 ) ) modified-Bessel-first-kind 𝜈 𝑧 1 2 superscript 𝑒 𝜈 𝜋 𝑖 2 Hankel-H-1-Bessel-third-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 2 Hankel-H-2-Bessel-third-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 2 {\displaystyle{\displaystyle I_{\nu}\left(z\right)=\tfrac{1}{2}e^{+\nu\pi i/2}% \left({H^{(1)}_{\nu}}\left(ze^{-\pi i/2}\right)+{H^{(2)}_{\nu}}\left(ze^{-\pi i% /2}\right)\right)}}
\modBesselI{\nu}@{z} = \tfrac{1}{2}e^{+\nu\pi i/2}\left(\HankelH{1}{\nu}@{ze^{-\pi i/2}}+\HankelH{2}{\nu}@{ze^{-\pi i/2}}\right)		 - π ≤ + ph ⁡ z , - π ≤ - ph ⁡ z , + ph ⁡ z ≤ 1 2 ⁢ π , - ph ⁡ z ≤ 1 2 ⁢ π , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜋 phase 𝑧 formulae-sequence 𝜋 phase 𝑧 formulae-sequence phase 𝑧 1 2 𝜋 formulae-sequence phase 𝑧 1 2 𝜋 𝜈 𝑘 1 0 {\displaystyle{\displaystyle-\pi\leq+\operatorname{ph}z,-\pi\leq-\operatorname% {ph}z,+\operatorname{ph}z\leq\tfrac{1}{2}\pi,-\operatorname{ph}z\leq\tfrac{1}{% 2}\pi,\Re(\nu+k+1)>0}} 
BesselI(nu, z) = (1)/(2)*exp(+ nu*Pi*I/2)*(HankelH1(nu, z*exp(- Pi*I/2))+ HankelH2(nu, z*exp(- Pi*I/2)))
 
BesselI[\[Nu], z] == Divide[1,2]*Exp[+ \[Nu]*Pi*I/2]*(HankelH1[\[Nu], z*Exp[- Pi*I/2]]+ HankelH2[\[Nu], z*Exp[- Pi*I/2]])
 Failure Failure Successful [Tested: 50] Successful [Tested: 50]
10.27.E9 π ⁢ i ⁢ J ν ⁡ ( z ) = e - ν ⁢ π ⁢ i / 2 ⁢ K ν ⁡ ( z ⁢ e - π ⁢ i / 2 ) - e ν ⁢ π ⁢ i / 2 ⁢ K ν ⁡ ( z ⁢ e π ⁢ i / 2 ) 𝜋 𝑖 Bessel-J 𝜈 𝑧 superscript 𝑒 𝜈 𝜋 𝑖 2 modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 2 superscript 𝑒 𝜈 𝜋 𝑖 2 modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 2 {\displaystyle{\displaystyle\pi iJ_{\nu}\left(z\right)=e^{-\nu\pi i/2}K_{\nu}% \left(ze^{-\pi i/2}\right)-e^{\nu\pi i/2}K_{\nu}\left(ze^{\pi i/2}\right)}}
\pi i\BesselJ{\nu}@{z} = e^{-\nu\pi i/2}\modBesselK{\nu}@{ze^{-\pi i/2}}-e^{\nu\pi i/2}\modBesselK{\nu}@{ze^{\pi i/2}}		 | ph ⁡ z | ≤ 1 2 ⁢ π , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence phase 𝑧 1 2 𝜋 𝜈 𝑘 1 0 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{1}{2}\pi,\Re(\nu+k+% 1)>0}} 
Pi*I*BesselJ(nu, z) = exp(- nu*Pi*I/2)*BesselK(nu, z*exp(- Pi*I/2))- exp(nu*Pi*I/2)*BesselK(nu, z*exp(Pi*I/2))
 
Pi*I*BesselJ[\[Nu], z] == Exp[- \[Nu]*Pi*I/2]*BesselK[\[Nu], z*Exp[- Pi*I/2]]- Exp[\[Nu]*Pi*I/2]*BesselK[\[Nu], z*Exp[Pi*I/2]]
 Failure Failure Successful [Tested: 50] Successful [Tested: 50]
10.27.E10 - π ⁢ Y ν ⁡ ( z ) = e - ν ⁢ π ⁢ i / 2 ⁢ K ν ⁡ ( z ⁢ e - π ⁢ i / 2 ) + e ν ⁢ π ⁢ i / 2 ⁢ K ν ⁡ ( z ⁢ e π ⁢ i / 2 ) 𝜋 Bessel-Y-Weber 𝜈 𝑧 superscript 𝑒 𝜈 𝜋 𝑖 2 modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 2 superscript 𝑒 𝜈 𝜋 𝑖 2 modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 2 {\displaystyle{\displaystyle-\pi Y_{\nu}\left(z\right)=e^{-\nu\pi i/2}K_{\nu}% \left(ze^{-\pi i/2}\right)+e^{\nu\pi i/2}K_{\nu}\left(ze^{\pi i/2}\right)}}
-\pi\BesselY{\nu}@{z} = e^{-\nu\pi i/2}\modBesselK{\nu}@{ze^{-\pi i/2}}+e^{\nu\pi i/2}\modBesselK{\nu}@{ze^{\pi i/2}}		 | ph ⁡ z | ≤ 1 2 ⁢ π , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 formulae-sequence phase 𝑧 1 2 𝜋 formulae-sequence 𝜈 𝑘 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{1}{2}\pi,\Re(\nu+k+% 1)>0,\Re((-\nu)+k+1)>0}} 
- Pi*BesselY(nu, z) = exp(- nu*Pi*I/2)*BesselK(nu, z*exp(- Pi*I/2))+ exp(nu*Pi*I/2)*BesselK(nu, z*exp(Pi*I/2))
 
- Pi*BesselY[\[Nu], z] == Exp[- \[Nu]*Pi*I/2]*BesselK[\[Nu], z*Exp[- Pi*I/2]]+ Exp[\[Nu]*Pi*I/2]*BesselK[\[Nu], z*Exp[Pi*I/2]]
 Failure Failure Successful [Tested: 50] Successful [Tested: 50]
10.27.E11 Y ν ⁡ ( z ) = e + ( ν + 1 ) ⁢ π ⁢ i / 2 ⁢ I ν ⁡ ( z ⁢ e - π ⁢ i / 2 ) - ( 2 / π ) ⁢ e - ν ⁢ π ⁢ i / 2 ⁢ K ν ⁡ ( z ⁢ e - π ⁢ i / 2 ) Bessel-Y-Weber 𝜈 𝑧 superscript 𝑒 𝜈 1 𝜋 𝑖 2 modified-Bessel-first-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 2 2 𝜋 superscript 𝑒 𝜈 𝜋 𝑖 2 modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 2 {\displaystyle{\displaystyle Y_{\nu}\left(z\right)=e^{+(\nu+1)\pi i/2}I_{\nu}% \left(ze^{-\pi i/2}\right)-(2/\pi)e^{-\nu\pi i/2}K_{\nu}\left(ze^{-\pi i/2}% \right)}}
\BesselY{\nu}@{z} = e^{+(\nu+1)\pi i/2}\modBesselI{\nu}@{ze^{-\pi i/2}}-(2/\pi)e^{-\nu\pi i/2}\modBesselK{\nu}@{ze^{-\pi i/2}}		 - 1 2 ⁢ π ≤ + ph ⁡ z , - 1 2 ⁢ π ≤ - ph ⁡ z , + ph ⁡ z ≤ π , - ph ⁡ z ≤ π , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 formulae-sequence 1 2 𝜋 phase 𝑧 formulae-sequence 1 2 𝜋 phase 𝑧 formulae-sequence phase 𝑧 𝜋 formulae-sequence phase 𝑧 𝜋 formulae-sequence 𝜈 𝑘 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle-\tfrac{1}{2}\pi\leq+\operatorname{ph}z,-\tfrac{1}% {2}\pi\leq-\operatorname{ph}z,+\operatorname{ph}z\leq\pi,-\operatorname{ph}z% \leq\pi,\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}} 
BesselY(nu, z) = exp(+(nu + 1)*Pi*I/2)*BesselI(nu, z*exp(- Pi*I/2))-(2/Pi)*exp(- nu*Pi*I/2)*BesselK(nu, z*exp(- Pi*I/2))
 
BesselY[\[Nu], z] == Exp[+(\[Nu]+ 1)*Pi*I/2]*BesselI[\[Nu], z*Exp[- Pi*I/2]]-(2/Pi)*Exp[- \[Nu]*Pi*I/2]*BesselK[\[Nu], z*Exp[- Pi*I/2]]
 Failure Failure Successful [Tested: 50] Successful [Tested: 50]
10.27.E11 Y ν ⁡ ( z ) = e - ( ν + 1 ) ⁢ π ⁢ i / 2 ⁢ I ν ⁡ ( z ⁢ e + π ⁢ i / 2 ) - ( 2 / π ) ⁢ e + ν ⁢ π ⁢ i / 2 ⁢ K ν ⁡ ( z ⁢ e + π ⁢ i / 2 ) Bessel-Y-Weber 𝜈 𝑧 superscript 𝑒 𝜈 1 𝜋 𝑖 2 modified-Bessel-first-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 2 2 𝜋 superscript 𝑒 𝜈 𝜋 𝑖 2 modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 2 {\displaystyle{\displaystyle Y_{\nu}\left(z\right)=e^{-(\nu+1)\pi i/2}I_{\nu}% \left(ze^{+\pi i/2}\right)-(2/\pi)e^{+\nu\pi i/2}K_{\nu}\left(ze^{+\pi i/2}% \right)}}
\BesselY{\nu}@{z} = e^{-(\nu+1)\pi i/2}\modBesselI{\nu}@{ze^{+\pi i/2}}-(2/\pi)e^{+\nu\pi i/2}\modBesselK{\nu}@{ze^{+\pi i/2}}		 - 1 2 ⁢ π ≤ + ph ⁡ z , - 1 2 ⁢ π ≤ - ph ⁡ z , + ph ⁡ z ≤ π , - ph ⁡ z ≤ π , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 formulae-sequence 1 2 𝜋 phase 𝑧 formulae-sequence 1 2 𝜋 phase 𝑧 formulae-sequence phase 𝑧 𝜋 formulae-sequence phase 𝑧 𝜋 formulae-sequence 𝜈 𝑘 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle-\tfrac{1}{2}\pi\leq+\operatorname{ph}z,-\tfrac{1}% {2}\pi\leq-\operatorname{ph}z,+\operatorname{ph}z\leq\pi,-\operatorname{ph}z% \leq\pi,\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}} 
BesselY(nu, z) = exp(-(nu + 1)*Pi*I/2)*BesselI(nu, z*exp(+ Pi*I/2))-(2/Pi)*exp(+ nu*Pi*I/2)*BesselK(nu, z*exp(+ Pi*I/2))
 
BesselY[\[Nu], z] == Exp[-(\[Nu]+ 1)*Pi*I/2]*BesselI[\[Nu], z*Exp[+ Pi*I/2]]-(2/Pi)*Exp[+ \[Nu]*Pi*I/2]*BesselK[\[Nu], z*Exp[+ Pi*I/2]]
 Failure Failure Successful [Tested: 50] Successful [Tested: 50]
10.28.E1 𝒲 ⁡ { I ν ⁡ ( z ) , I - ν ⁡ ( z ) } = I ν ⁡ ( z ) ⁢ I - ν - 1 ⁡ ( z ) - I ν + 1 ⁡ ( z ) ⁢ I - ν ⁡ ( z ) Wronskian modified-Bessel-first-kind 𝜈 𝑧 modified-Bessel-first-kind 𝜈 𝑧 modified-Bessel-first-kind 𝜈 𝑧 modified-Bessel-first-kind 𝜈 1 𝑧 modified-Bessel-first-kind 𝜈 1 𝑧 modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle\mathscr{W}\left\{I_{\nu}\left(z\right),I_{-\nu}% \left(z\right)\right\}=I_{\nu}\left(z\right)I_{-\nu-1}\left(z\right)-I_{\nu+1}% \left(z\right)I_{-\nu}\left(z\right)}}
\Wronskian@{\modBesselI{\nu}@{z},\modBesselI{-\nu}@{z}} = \modBesselI{\nu}@{z}\modBesselI{-\nu-1}@{z}-\modBesselI{\nu+1}@{z}\modBesselI{-\nu}@{z}		 ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ν - 1 ) + k + 1 ) > 0 , ℜ ⁡ ( ( ν + 1 ) + k + 1 ) > 0 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 𝜈 1 𝑘 1 0 𝜈 1 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re((-\nu-1)+k+1)% >0,\Re((\nu+1)+k+1)>0}} 
(BesselI(nu, z))*diff(BesselI(- nu, z), z)-diff(BesselI(nu, z), z)*(BesselI(- nu, z)) = BesselI(nu, z)*BesselI(- nu - 1, z)- BesselI(nu + 1, z)*BesselI(- nu, z)
 
Wronskian[{BesselI[\[Nu], z], BesselI[- \[Nu], z]}, z] == BesselI[\[Nu], z]*BesselI[- \[Nu]- 1, z]- BesselI[\[Nu]+ 1, z]*BesselI[- \[Nu], z]
 Successful Successful Skip - symbolical successful subtest Successful [Tested: 70]
10.28.E1 I ν ⁡ ( z ) ⁢ I - ν - 1 ⁡ ( z ) - I ν + 1 ⁡ ( z ) ⁢ I - ν ⁡ ( z ) = - 2 ⁢ sin ⁡ ( ν ⁢ π ) / ( π ⁢ z ) modified-Bessel-first-kind 𝜈 𝑧 modified-Bessel-first-kind 𝜈 1 𝑧 modified-Bessel-first-kind 𝜈 1 𝑧 modified-Bessel-first-kind 𝜈 𝑧 2 𝜈 𝜋 𝜋 𝑧 {\displaystyle{\displaystyle I_{\nu}\left(z\right)I_{-\nu-1}\left(z\right)-I_{% \nu+1}\left(z\right)I_{-\nu}\left(z\right)=-2\sin\left(\nu\pi\right)/(\pi z)}}
\modBesselI{\nu}@{z}\modBesselI{-\nu-1}@{z}-\modBesselI{\nu+1}@{z}\modBesselI{-\nu}@{z} = -2\sin@{\nu\pi}/(\pi z)		 ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ν - 1 ) + k + 1 ) > 0 , ℜ ⁡ ( ( ν + 1 ) + k + 1 ) > 0 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 𝜈 1 𝑘 1 0 𝜈 1 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re((-\nu-1)+k+1)% >0,\Re((\nu+1)+k+1)>0}} 
BesselI(nu, z)*BesselI(- nu - 1, z)- BesselI(nu + 1, z)*BesselI(- nu, z) = - 2*sin(nu*Pi)/(Pi*z)
 
BesselI[\[Nu], z]*BesselI[- \[Nu]- 1, z]- BesselI[\[Nu]+ 1, z]*BesselI[- \[Nu], z] == - 2*Sin[\[Nu]*Pi]/(Pi*z)
 Failure Successful Successful [Tested: 70] Successful [Tested: 70]
10.28.E2 𝒲 ⁡ { K ν ⁡ ( z ) , I ν ⁡ ( z ) } = I ν ⁡ ( z ) ⁢ K ν + 1 ⁡ ( z ) + I ν + 1 ⁡ ( z ) ⁢ K ν ⁡ ( z ) Wronskian modified-Bessel-second-kind 𝜈 𝑧 modified-Bessel-first-kind 𝜈 𝑧 modified-Bessel-first-kind 𝜈 𝑧 modified-Bessel-second-kind 𝜈 1 𝑧 modified-Bessel-first-kind 𝜈 1 𝑧 modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle\mathscr{W}\left\{K_{\nu}\left(z\right),I_{\nu}% \left(z\right)\right\}=I_{\nu}\left(z\right)K_{\nu+1}\left(z\right)+I_{\nu+1}% \left(z\right)K_{\nu}\left(z\right)}}
\Wronskian@{\modBesselK{\nu}@{z},\modBesselI{\nu}@{z}} = \modBesselI{\nu}@{z}\modBesselK{\nu+1}@{z}+\modBesselI{\nu+1}@{z}\modBesselK{\nu}@{z}		 ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( ν + 1 ) + k + 1 ) > 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 1 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((\nu+1)+k+1)>0}} 
(BesselK(nu, z))*diff(BesselI(nu, z), z)-diff(BesselK(nu, z), z)*(BesselI(nu, z)) = BesselI(nu, z)*BesselK(nu + 1, z)+ BesselI(nu + 1, z)*BesselK(nu, z)
 
Wronskian[{BesselK[\[Nu], z], BesselI[\[Nu], z]}, z] == BesselI[\[Nu], z]*BesselK[\[Nu]+ 1, z]+ BesselI[\[Nu]+ 1, z]*BesselK[\[Nu], z]
 Successful Successful Skip - symbolical successful subtest Successful [Tested: 70]
10.28.E2 I ν ⁡ ( z ) ⁢ K ν + 1 ⁡ ( z ) + I ν + 1 ⁡ ( z ) ⁢ K ν ⁡ ( z ) = 1 / z modified-Bessel-first-kind 𝜈 𝑧 modified-Bessel-second-kind 𝜈 1 𝑧 modified-Bessel-first-kind 𝜈 1 𝑧 modified-Bessel-second-kind 𝜈 𝑧 1 𝑧 {\displaystyle{\displaystyle I_{\nu}\left(z\right)K_{\nu+1}\left(z\right)+I_{% \nu+1}\left(z\right)K_{\nu}\left(z\right)=1/z}}
\modBesselI{\nu}@{z}\modBesselK{\nu+1}@{z}+\modBesselI{\nu+1}@{z}\modBesselK{\nu}@{z} = 1/z		 ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( ν + 1 ) + k + 1 ) > 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 1 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((\nu+1)+k+1)>0}} 
BesselI(nu, z)*BesselK(nu + 1, z)+ BesselI(nu + 1, z)*BesselK(nu, z) = 1/z
 
BesselI[\[Nu], z]*BesselK[\[Nu]+ 1, z]+ BesselI[\[Nu]+ 1, z]*BesselK[\[Nu], z] == 1/z
 Failure Successful Successful [Tested: 70] Successful [Tested: 70]
10.29#Ex5 I 0 ′ ⁡ ( z ) = I 1 ⁡ ( z ) diffop modified-Bessel-first-kind 0 1 𝑧 modified-Bessel-first-kind 1 𝑧 {\displaystyle{\displaystyle I_{0}'\left(z\right)=I_{1}\left(z\right)}}
\modBesselI{0}'@{z} = \modBesselI{1}@{z}		 ℜ ⁡ ( 0 + k + 1 ) > 0 , ℜ ⁡ ( 1 + k + 1 ) > 0 formulae-sequence 0 𝑘 1 0 1 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(1+k+1)>0}} 
diff( BesselI(0, z), z$(1) ) = BesselI(1, z)
 
D[BesselI[0, z], {z, 1}] == BesselI[1, z]
 Successful Successful - Successful [Tested: 7]
10.29#Ex6 K 0 ′ ⁡ ( z ) = - K 1 ⁡ ( z ) diffop modified-Bessel-second-kind 0 1 𝑧 modified-Bessel-second-kind 1 𝑧 {\displaystyle{\displaystyle K_{0}'\left(z\right)=-K_{1}\left(z\right)}}
\modBesselK{0}'@{z} = -\modBesselK{1}@{z}		 

diff( BesselK(0, z), z$(1) ) = - BesselK(1, z)
 
D[BesselK[0, z], {z, 1}] == - BesselK[1, z]
 Successful Successful - Successful [Tested: 7]
10.31.E1 K n ⁡ ( z ) = 1 2 ⁢ ( 1 2 ⁢ z ) - n ⁢ ∑ k = 0 n - 1 ( n - k - 1 ) ! k ! ⁢ ( - 1 4 ⁢ z 2 ) k + ( - 1 ) n + 1 ⁢ ln ⁡ ( 1 2 ⁢ z ) ⁢ I n ⁡ ( z ) + ( - 1 ) n ⁢ 1 2 ⁢ ( 1 2 ⁢ z ) n ⁢ ∑ k = 0 ∞ ( ψ ⁡ ( k + 1 ) + ψ ⁡ ( n + k + 1 ) ) ⁢ ( 1 4 ⁢ z 2 ) k k ! ⁢ ( n + k ) ! modified-Bessel-second-kind 𝑛 𝑧 1 2 superscript 1 2 𝑧 𝑛 superscript subscript 𝑘 0 𝑛 1 𝑛 𝑘 1 𝑘 superscript 1 4 superscript 𝑧 2 𝑘 superscript 1 𝑛 1 1 2 𝑧 modified-Bessel-first-kind 𝑛 𝑧 superscript 1 𝑛 1 2 superscript 1 2 𝑧 𝑛 superscript subscript 𝑘 0 digamma 𝑘 1 digamma 𝑛 𝑘 1 superscript 1 4 superscript 𝑧 2 𝑘 𝑘 𝑛 𝑘 {\displaystyle{\displaystyle K_{n}\left(z\right)=\tfrac{1}{2}(\tfrac{1}{2}z)^{% -n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}(-\tfrac{1}{4}z^{2})^{k}+(-1)^{n+1}\ln% \left(\tfrac{1}{2}z\right)I_{n}\left(z\right)+(-1)^{n}\tfrac{1}{2}(\tfrac{1}{2% }z)^{n}\sum_{k=0}^{\infty}\left(\psi\left(k+1\right)+\psi\left(n+k+1\right)% \right)\frac{(\tfrac{1}{4}z^{2})^{k}}{k!(n+k)!}}}
\modBesselK{n}@{z} = \tfrac{1}{2}(\tfrac{1}{2}z)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}(-\tfrac{1}{4}z^{2})^{k}+(-1)^{n+1}\ln@{\tfrac{1}{2}z}\modBesselI{n}@{z}+(-1)^{n}\tfrac{1}{2}(\tfrac{1}{2}z)^{n}\sum_{k=0}^{\infty}\left(\digamma@{k+1}+\digamma@{n+k+1}\right)\frac{(\tfrac{1}{4}z^{2})^{k}}{k!(n+k)!}		 ℜ ⁡ ( n + k + 1 ) > 0 𝑛 𝑘 1 0 {\displaystyle{\displaystyle\Re(n+k+1)>0}} 
BesselK(n, z) = (1)/(2)*((1)/(2)*z)^(- n)* sum((factorial(n - k - 1))/(factorial(k))*(-(1)/(4)*(z)^(2))^(k), k = 0..n - 1)+(- 1)^(n + 1)* ln((1)/(2)*z)*BesselI(n, z)+(- 1)^(n)*(1)/(2)*((1)/(2)*z)^(n)* sum((Psi(k + 1)+ Psi(n + k + 1))*(((1)/(4)*(z)^(2))^(k))/(factorial(k)*factorial(n + k)), k = 0..infinity)
 
BesselK[n, z] == Divide[1,2]*(Divide[1,2]*z)^(- n)* Sum[Divide[(n - k - 1)!,(k)!]*(-Divide[1,4]*(z)^(2))^(k), {k, 0, n - 1}, GenerateConditions->None]+(- 1)^(n + 1)* Log[Divide[1,2]*z]*BesselI[n, z]+(- 1)^(n)*Divide[1,2]*(Divide[1,2]*z)^(n)* Sum[(PolyGamma[k + 1]+ PolyGamma[n + k + 1])*Divide[(Divide[1,4]*(z)^(2))^(k),(k)!*(n + k)!], {k, 0, Infinity}, GenerateConditions->None]
 Aborted Aborted Skipped - Because timed out
Failed [6 / 21]
Result: Plus[0.6666666666666666, Times[-0.6666666666666666, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-4, []], Times[Plus[12, Times[8, ]], [Plus[1, ]]], Times[Plus[-16, Times[-16, ], Times[-4, Power[, 2]], Power[1.5, 2]], [Plus[2, ]]], Times[-1, Plus[2, ], Power[1.5, 2], [Plus[3, ]]]], 0], Equal[[1], 1], Equal[[2], Plus[1, Times[-4, Power[1.5, -2]]]], Equal[[3], Plus[Rational[1, 2], Times[16, Power[1.5, -4], Plus[2, Times[Rational[-1, 4], Power[1.5, 2]]]]]], Equal[[4], Times[Rational[-32, 3], Power[1.5, -6], Plus[3, Times[Rational[-1, 4], Power[1.5, 2]]], Plus[12, Times[Rational[1, 16], Power[1.5, 4]]]]]}]][1.0]]], {Rule[n, 1], Rule[z, 1.5]}

Result: Plus[0.38888888888888906, Times[0.5, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-4, []], Times[Plus[12, Times[8, ]], [Plus[1, ]]], Times[Plus[-16, Times[-16, ], Times[-4, Power[, 2]], Power[1.5, 2]], [Plus[2, ]]], Times[-1, Plus[2, ], Power[1.5, 2], [Plus[3, ]]]], 0], Equal[[1], 1], Equal[[2], Plus[1, Times[-4, Power[1.5, -2]]]], Equal[[3], Plus[Rational[1, 2], Times[16, Power[1.5, -4], Plus[2, Times[Rational[-1, 4], Power[1.5, 2]]]]]], Equal[[4], Times[Rational[-32, 3], Power[1.5, -6], Plus[3, Times[Rational[-1, 4], Power[1.5, 2]]], Plus[12, Times[Rational[1, 16], Power[1.5, 4]]]]]}]][2.0]]], {Rule[n, 2], Rule[z, 1.5]}

... skip entries to safe data
10.31.E2 K 0 ⁡ ( z ) = - ( ln ⁡ ( 1 2 ⁢ z ) + γ ) ⁢ I 0 ⁡ ( z ) + 1 4 ⁢ z 2 ( 1 ! ) 2 + ( 1 + 1 2 ) ⁢ ( 1 4 ⁢ z 2 ) 2 ( 2 ! ) 2 + ( 1 + 1 2 + 1 3 ) ⁢ ( 1 4 ⁢ z 2 ) 3 ( 3 ! ) 2 + ⋯ modified-Bessel-second-kind 0 𝑧 1 2 𝑧 modified-Bessel-first-kind 0 𝑧 1 4 superscript 𝑧 2 superscript 1 2 1 1 2 superscript 1 4 superscript 𝑧 2 2 superscript 2 2 1 1 2 1 3 superscript 1 4 superscript 𝑧 2 3 superscript 3 2 ⋯ {\displaystyle{\displaystyle K_{0}\left(z\right)=-\left(\ln\left(\tfrac{1}{2}z% \right)+\gamma\right)I_{0}\left(z\right)+\frac{\tfrac{1}{4}z^{2}}{(1!)^{2}}+(1% +\tfrac{1}{2})\frac{(\tfrac{1}{4}z^{2})^{2}}{(2!)^{2}}+(1+\tfrac{1}{2}+\tfrac{% 1}{3})\frac{(\tfrac{1}{4}z^{2})^{3}}{(3!)^{2}}+\cdots}}
\modBesselK{0}@{z} = -\left(\ln@{\tfrac{1}{2}z}+\EulerConstant\right)\modBesselI{0}@{z}+\frac{\tfrac{1}{4}z^{2}}{(1!)^{2}}+(1+\tfrac{1}{2})\frac{(\tfrac{1}{4}z^{2})^{2}}{(2!)^{2}}+(1+\tfrac{1}{2}+\tfrac{1}{3})\frac{(\tfrac{1}{4}z^{2})^{3}}{(3!)^{2}}+\dotsi		 ℜ ⁡ ( 0 + k + 1 ) > 0 0 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0}} 
BesselK(0, z) = -(ln((1)/(2)*z)+ gamma)*BesselI(0, z)+((1)/(4)*(z)^(2))/((factorial(1))^(2))+(1 +(1)/(2))*(((1)/(4)*(z)^(2))^(2))/((factorial(2))^(2))+(1 +(1)/(2)+(1)/(3))*(((1)/(4)*(z)^(2))^(3))/((factorial(3))^(2))+ ..
 
BesselK[0, z] == -(Log[Divide[1,2]*z]+ EulerGamma)*BesselI[0, z]+Divide[Divide[1,4]*(z)^(2),((1)!)^(2)]+(1 +Divide[1,2])*Divide[(Divide[1,4]*(z)^(2))^(2),((2)!)^(2)]+(1 +Divide[1,2]+Divide[1,3])*Divide[(Divide[1,4]*(z)^(2))^(3),((3)!)^(2)]+ \[Ellipsis]
 Error Failure -
Failed [7 / 7]
Result: Plus[Complex[-6.985673039111573*^-6, -1.2369744460005716*^-5], Times[-1.0, …]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Plus[Complex[-7.140527721077872*^-6, -1.2101549865001227*^-5], Times[-1.0, …]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.31.E3 I ν ⁡ ( z ) ⁢ I μ ⁡ ( z ) = ( 1 2 ⁢ z ) ν + μ ⁢ ∑ k = 0 ∞ ( ν + μ + k + 1 ) k ⁢ ( 1 4 ⁢ z 2 ) k k ! ⁢ Γ ⁡ ( ν + k + 1 ) ⁢ Γ ⁡ ( μ + k + 1 ) modified-Bessel-first-kind 𝜈 𝑧 modified-Bessel-first-kind 𝜇 𝑧 superscript 1 2 𝑧 𝜈 𝜇 superscript subscript 𝑘 0 subscript 𝜈 𝜇 𝑘 1 𝑘 superscript 1 4 superscript 𝑧 2 𝑘 𝑘 Euler-Gamma 𝜈 𝑘 1 Euler-Gamma 𝜇 𝑘 1 {\displaystyle{\displaystyle I_{\nu}\left(z\right)I_{\mu}\left(z\right)=(% \tfrac{1}{2}z)^{\nu+\mu}\sum_{k=0}^{\infty}\frac{(\nu+\mu+k+1)_{k}(\tfrac{1}{4% }z^{2})^{k}}{k!\Gamma\left(\nu+k+1\right)\Gamma\left(\mu+k+1\right)}}}
\modBesselI{\nu}@{z}\modBesselI{\mu}@{z} = (\tfrac{1}{2}z)^{\nu+\mu}\sum_{k=0}^{\infty}\frac{(\nu+\mu+k+1)_{k}(\tfrac{1}{4}z^{2})^{k}}{k!\EulerGamma@{\nu+k+1}\EulerGamma@{\mu+k+1}}		 ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( μ + k + 1 ) > 0 , ℜ ⁡ ( ( μ ) + k + 1 ) > 0 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 𝜇 𝑘 1 0 𝜇 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\mu+k+1)>0,\Re((\mu)+k+1)>0}} 
BesselI(nu, z)*BesselI(mu, z) = ((1)/(2)*z)^(nu + mu)* sum((nu + mu + k + 1[k]*((1)/(4)*(z)^(2))^(k))/(factorial(k)*GAMMA(nu + k + 1)*GAMMA(mu + k + 1)), k = 0..infinity)
 
BesselI[\[Nu], z]*BesselI[\[Mu], z] == (Divide[1,2]*z)^(\[Nu]+ \[Mu])* Sum[Divide[Subscript[\[Nu]+ \[Mu]+ k + 1, k]*(Divide[1,4]*(z)^(2))^(k),(k)!*Gamma[\[Nu]+ k + 1]*Gamma[\[Mu]+ k + 1]], {k, 0, Infinity}, GenerateConditions->None]
 Failure Failure Skipped - Because timed out Skipped - Because timed out
10.32.E1 I 0 ⁡ ( z ) = 1 π ⁢ ∫ 0 π e + z ⁢ cos ⁡ θ ⁢ d θ modified-Bessel-first-kind 0 𝑧 1 𝜋 superscript subscript 0 𝜋 superscript 𝑒 𝑧 𝜃 𝜃 {\displaystyle{\displaystyle I_{0}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}e^% {+z\cos\theta}\mathrm{d}\theta}}
\modBesselI{0}@{z} = \frac{1}{\pi}\int_{0}^{\pi}e^{+ z\cos@@{\theta}}\diff{\theta}		 ℜ ⁡ ( 0 + k + 1 ) > 0 0 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0}} 
BesselI(0, z) = (1)/(Pi)*int(exp(+ z*cos(theta)), theta = 0..Pi)
 
BesselI[0, z] == Divide[1,Pi]*Integrate[Exp[+ z*Cos[\[Theta]]], {\[Theta], 0, Pi}, GenerateConditions->None]
 Successful Successful Skip - symbolical successful subtest Successful [Tested: 7]
10.32.E1 I 0 ⁡ ( z ) = 1 π ⁢ ∫ 0 π e - z ⁢ cos ⁡ θ ⁢ d θ modified-Bessel-first-kind 0 𝑧 1 𝜋 superscript subscript 0 𝜋 superscript 𝑒 𝑧 𝜃 𝜃 {\displaystyle{\displaystyle I_{0}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}e^% {-z\cos\theta}\mathrm{d}\theta}}
\modBesselI{0}@{z} = \frac{1}{\pi}\int_{0}^{\pi}e^{- z\cos@@{\theta}}\diff{\theta}		 ℜ ⁡ ( 0 + k + 1 ) > 0 0 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0}} 
BesselI(0, z) = (1)/(Pi)*int(exp(- z*cos(theta)), theta = 0..Pi)
 
BesselI[0, z] == Divide[1,Pi]*Integrate[Exp[- z*Cos[\[Theta]]], {\[Theta], 0, Pi}, GenerateConditions->None]
 Successful Successful Skip - symbolical successful subtest Successful [Tested: 7]
10.32.E1 1 π ⁢ ∫ 0 π e + z ⁢ cos ⁡ θ ⁢ d θ = 1 π ⁢ ∫ 0 π cosh ⁡ ( z ⁢ cos ⁡ θ ) ⁢ d θ 1 𝜋 superscript subscript 0 𝜋 superscript 𝑒 𝑧 𝜃 𝜃 1 𝜋 superscript subscript 0 𝜋 𝑧 𝜃 𝜃 {\displaystyle{\displaystyle\frac{1}{\pi}\int_{0}^{\pi}e^{+z\cos\theta}\mathrm% {d}\theta=\frac{1}{\pi}\int_{0}^{\pi}\cosh\left(z\cos\theta\right)\mathrm{d}% \theta}}
\frac{1}{\pi}\int_{0}^{\pi}e^{+ z\cos@@{\theta}}\diff{\theta} = \frac{1}{\pi}\int_{0}^{\pi}\cosh@{z\cos@@{\theta}}\diff{\theta}		 ℜ ⁡ ( 0 + k + 1 ) > 0 0 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0}} 
(1)/(Pi)*int(exp(+ z*cos(theta)), theta = 0..Pi) = (1)/(Pi)*int(cosh(z*cos(theta)), theta = 0..Pi)
 
Divide[1,Pi]*Integrate[Exp[+ z*Cos[\[Theta]]], {\[Theta], 0, Pi}, GenerateConditions->None] == Divide[1,Pi]*Integrate[Cosh[z*Cos[\[Theta]]], {\[Theta], 0, Pi}, GenerateConditions->None]
 Failure Failure Skipped - Because timed out Successful [Tested: 7]
10.32.E1 1 π ⁢ ∫ 0 π e - z ⁢ cos ⁡ θ ⁢ d θ = 1 π ⁢ ∫ 0 π cosh ⁡ ( z ⁢ cos ⁡ θ ) ⁢ d θ 1 𝜋 superscript subscript 0 𝜋 superscript 𝑒 𝑧 𝜃 𝜃 1 𝜋 superscript subscript 0 𝜋 𝑧 𝜃 𝜃 {\displaystyle{\displaystyle\frac{1}{\pi}\int_{0}^{\pi}e^{-z\cos\theta}\mathrm% {d}\theta=\frac{1}{\pi}\int_{0}^{\pi}\cosh\left(z\cos\theta\right)\mathrm{d}% \theta}}
\frac{1}{\pi}\int_{0}^{\pi}e^{- z\cos@@{\theta}}\diff{\theta} = \frac{1}{\pi}\int_{0}^{\pi}\cosh@{z\cos@@{\theta}}\diff{\theta}		 ℜ ⁡ ( 0 + k + 1 ) > 0 0 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0}} 
(1)/(Pi)*int(exp(- z*cos(theta)), theta = 0..Pi) = (1)/(Pi)*int(cosh(z*cos(theta)), theta = 0..Pi)
 
Divide[1,Pi]*Integrate[Exp[- z*Cos[\[Theta]]], {\[Theta], 0, Pi}, GenerateConditions->None] == Divide[1,Pi]*Integrate[Cosh[z*Cos[\[Theta]]], {\[Theta], 0, Pi}, GenerateConditions->None]
 Failure Failure Skipped - Because timed out Successful [Tested: 7]
10.32.E2 I ν ⁡ ( z ) = ( 1 2 ⁢ z ) ν π 1 2 ⁢ Γ ⁡ ( ν + 1 2 ) ⁢ ∫ 0 π e + z ⁢ cos ⁡ θ ⁢ ( sin ⁡ θ ) 2 ⁢ ν ⁢ d θ modified-Bessel-first-kind 𝜈 𝑧 superscript 1 2 𝑧 𝜈 superscript 𝜋 1 2 Euler-Gamma 𝜈 1 2 superscript subscript 0 𝜋 superscript 𝑒 𝑧 𝜃 superscript 𝜃 2 𝜈 𝜃 {\displaystyle{\displaystyle I_{\nu}\left(z\right)=\frac{(\frac{1}{2}z)^{\nu}}% {\pi^{\frac{1}{2}}\Gamma\left(\nu+\frac{1}{2}\right)}\int_{0}^{\pi}e^{+z\cos% \theta}(\sin\theta)^{2\nu}\mathrm{d}\theta}}
\modBesselI{\nu}@{z} = \frac{(\frac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\nu+\frac{1}{2}}}\int_{0}^{\pi}e^{+ z\cos@@{\theta}}(\sin@@{\theta})^{2\nu}\diff{\theta}		 ℜ ⁡ ν > - 1 2 , ℜ ⁡ ( ν + 1 2 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence 𝜈 1 2 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+\frac{1}{2})>0,\Re(% \nu+k+1)>0}} 
BesselI(nu, z) = (((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int(exp(+ z*cos(theta))*(sin(theta))^(2*nu), theta = 0..Pi)
 
BesselI[\[Nu], z] == Divide[(Divide[1,2]*z)^\[Nu],(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[Exp[+ z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*\[Nu]), {\[Theta], 0, Pi}, GenerateConditions->None]
 Failure Aborted Skipped - Because timed out Successful [Tested: 35]
10.32.E2 I ν ⁡ ( z ) = ( 1 2 ⁢ z ) ν π 1 2 ⁢ Γ ⁡ ( ν + 1 2 ) ⁢ ∫ 0 π e - z ⁢ cos ⁡ θ ⁢ ( sin ⁡ θ ) 2 ⁢ ν ⁢ d θ modified-Bessel-first-kind 𝜈 𝑧 superscript 1 2 𝑧 𝜈 superscript 𝜋 1 2 Euler-Gamma 𝜈 1 2 superscript subscript 0 𝜋 superscript 𝑒 𝑧 𝜃 superscript 𝜃 2 𝜈 𝜃 {\displaystyle{\displaystyle I_{\nu}\left(z\right)=\frac{(\frac{1}{2}z)^{\nu}}% {\pi^{\frac{1}{2}}\Gamma\left(\nu+\frac{1}{2}\right)}\int_{0}^{\pi}e^{-z\cos% \theta}(\sin\theta)^{2\nu}\mathrm{d}\theta}}
\modBesselI{\nu}@{z} = \frac{(\frac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\nu+\frac{1}{2}}}\int_{0}^{\pi}e^{- z\cos@@{\theta}}(\sin@@{\theta})^{2\nu}\diff{\theta}		 ℜ ⁡ ν > - 1 2 , ℜ ⁡ ( ν + 1 2 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence 𝜈 1 2 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+\frac{1}{2})>0,\Re(% \nu+k+1)>0}} 
BesselI(nu, z) = (((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int(exp(- z*cos(theta))*(sin(theta))^(2*nu), theta = 0..Pi)
 
BesselI[\[Nu], z] == Divide[(Divide[1,2]*z)^\[Nu],(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[Exp[- z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*\[Nu]), {\[Theta], 0, Pi}, GenerateConditions->None]
 Failure Aborted Skipped - Because timed out Successful [Tested: 35]
10.32.E2 ( 1 2 ⁢ z ) ν π 1 2 ⁢ Γ ⁡ ( ν + 1 2 ) ⁢ ∫ 0 π e + z ⁢ cos ⁡ θ ⁢ ( sin ⁡ θ ) 2 ⁢ ν ⁢ d θ = ( 1 2 ⁢ z ) ν π 1 2 ⁢ Γ ⁡ ( ν + 1 2 ) ⁢ ∫ - 1 1 ( 1 - t 2 ) ν - 1 2 ⁢ e + z ⁢ t ⁢ d t superscript 1 2 𝑧 𝜈 superscript 𝜋 1 2 Euler-Gamma 𝜈 1 2 superscript subscript 0 𝜋 superscript 𝑒 𝑧 𝜃 superscript 𝜃 2 𝜈 𝜃 superscript 1 2 𝑧 𝜈 superscript 𝜋 1 2 Euler-Gamma 𝜈 1 2 superscript subscript 1 1 superscript 1 superscript 𝑡 2 𝜈 1 2 superscript 𝑒 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle\frac{(\frac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}% \Gamma\left(\nu+\frac{1}{2}\right)}\int_{0}^{\pi}e^{+z\cos\theta}(\sin\theta)^% {2\nu}\mathrm{d}\theta=\frac{(\frac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\Gamma% \left(\nu+\frac{1}{2}\right)}\int_{-1}^{1}(1-t^{2})^{\nu-\frac{1}{2}}e^{+zt}% \mathrm{d}t}}
\frac{(\frac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\nu+\frac{1}{2}}}\int_{0}^{\pi}e^{+ z\cos@@{\theta}}(\sin@@{\theta})^{2\nu}\diff{\theta} = \frac{(\frac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\nu+\frac{1}{2}}}\int_{-1}^{1}(1-t^{2})^{\nu-\frac{1}{2}}e^{+ zt}\diff{t}		 ℜ ⁡ ν > - 1 2 , ℜ ⁡ ( ν + 1 2 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence 𝜈 1 2 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+\frac{1}{2})>0,\Re(% \nu+k+1)>0}} 
(((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int(exp(+ z*cos(theta))*(sin(theta))^(2*nu), theta = 0..Pi) = (((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int((1 - (t)^(2))^(nu -(1)/(2))* exp(+ z*t), t = - 1..1)
 
Divide[(Divide[1,2]*z)^\[Nu],(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[Exp[+ z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*\[Nu]), {\[Theta], 0, Pi}, GenerateConditions->None] == Divide[(Divide[1,2]*z)^\[Nu],(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[(1 - (t)^(2))^(\[Nu]-Divide[1,2])* Exp[+ z*t], {t, - 1, 1}, GenerateConditions->None]
 Failure Aborted Skipped - Because timed out Successful [Tested: 35]
10.32.E2 ( 1 2 ⁢ z ) ν π 1 2 ⁢ Γ ⁡ ( ν + 1 2 ) ⁢ ∫ 0 π e - z ⁢ cos ⁡ θ ⁢ ( sin ⁡ θ ) 2 ⁢ ν ⁢ d θ = ( 1 2 ⁢ z ) ν π 1 2 ⁢ Γ ⁡ ( ν + 1 2 ) ⁢ ∫ - 1 1 ( 1 - t 2 ) ν - 1 2 ⁢ e - z ⁢ t ⁢ d t superscript 1 2 𝑧 𝜈 superscript 𝜋 1 2 Euler-Gamma 𝜈 1 2 superscript subscript 0 𝜋 superscript 𝑒 𝑧 𝜃 superscript 𝜃 2 𝜈 𝜃 superscript 1 2 𝑧 𝜈 superscript 𝜋 1 2 Euler-Gamma 𝜈 1 2 superscript subscript 1 1 superscript 1 superscript 𝑡 2 𝜈 1 2 superscript 𝑒 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle\frac{(\frac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}% \Gamma\left(\nu+\frac{1}{2}\right)}\int_{0}^{\pi}e^{-z\cos\theta}(\sin\theta)^% {2\nu}\mathrm{d}\theta=\frac{(\frac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\Gamma% \left(\nu+\frac{1}{2}\right)}\int_{-1}^{1}(1-t^{2})^{\nu-\frac{1}{2}}e^{-zt}% \mathrm{d}t}}
\frac{(\frac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\nu+\frac{1}{2}}}\int_{0}^{\pi}e^{- z\cos@@{\theta}}(\sin@@{\theta})^{2\nu}\diff{\theta} = \frac{(\frac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\nu+\frac{1}{2}}}\int_{-1}^{1}(1-t^{2})^{\nu-\frac{1}{2}}e^{- zt}\diff{t}		 ℜ ⁡ ν > - 1 2 , ℜ ⁡ ( ν + 1 2 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence 𝜈 1 2 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+\frac{1}{2})>0,\Re(% \nu+k+1)>0}} 
(((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int(exp(- z*cos(theta))*(sin(theta))^(2*nu), theta = 0..Pi) = (((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int((1 - (t)^(2))^(nu -(1)/(2))* exp(- z*t), t = - 1..1)
 
Divide[(Divide[1,2]*z)^\[Nu],(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[Exp[- z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*\[Nu]), {\[Theta], 0, Pi}, GenerateConditions->None] == Divide[(Divide[1,2]*z)^\[Nu],(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[(1 - (t)^(2))^(\[Nu]-Divide[1,2])* Exp[- z*t], {t, - 1, 1}, GenerateConditions->None]
 Error Aborted Skip - symbolical successful subtest Successful [Tested: 35]
10.32.E3 I n ⁡ ( z ) = 1 π ⁢ ∫ 0 π e z ⁢ cos ⁡ θ ⁢ cos ⁡ ( n ⁢ θ ) ⁢ d θ modified-Bessel-first-kind 𝑛 𝑧 1 𝜋 superscript subscript 0 𝜋 superscript 𝑒 𝑧 𝜃 𝑛 𝜃 𝜃 {\displaystyle{\displaystyle I_{n}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}e^% {z\cos\theta}\cos\left(n\theta\right)\mathrm{d}\theta}}
\modBesselI{n}@{z} = \frac{1}{\pi}\int_{0}^{\pi}e^{z\cos@@{\theta}}\cos@{n\theta}\diff{\theta}		 ℜ ⁡ ( n + k + 1 ) > 0 𝑛 𝑘 1 0 {\displaystyle{\displaystyle\Re(n+k+1)>0}} 
BesselI(n, z) = (1)/(Pi)*int(exp(z*cos(theta))*cos(n*theta), theta = 0..Pi)
 
BesselI[n, z] == Divide[1,Pi]*Integrate[Exp[z*Cos[\[Theta]]]*Cos[n*\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None]
 Failure Aborted Successful [Tested: 21] Skipped - Because timed out
10.32.E4 I ν ⁡ ( z ) = 1 π ⁢ ∫ 0 π e z ⁢ cos ⁡ θ ⁢ cos ⁡ ( ν ⁢ θ ) ⁢ d θ - sin ⁡ ( ν ⁢ π ) π ⁢ ∫ 0 ∞ e - z ⁢ cosh ⁡ t - ν ⁢ t ⁢ d t modified-Bessel-first-kind 𝜈 𝑧 1 𝜋 superscript subscript 0 𝜋 superscript 𝑒 𝑧 𝜃 𝜈 𝜃 𝜃 𝜈 𝜋 𝜋 superscript subscript 0 superscript 𝑒 𝑧 𝑡 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle I_{\nu}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}% e^{z\cos\theta}\cos\left(\nu\theta\right)\mathrm{d}\theta-\frac{\sin\left(\nu% \pi\right)}{\pi}\int_{0}^{\infty}e^{-z\cosh t-\nu t}\mathrm{d}t}}
\modBesselI{\nu}@{z} = \frac{1}{\pi}\int_{0}^{\pi}e^{z\cos@@{\theta}}\cos@{\nu\theta}\diff{\theta}-\frac{\sin@{\nu\pi}}{\pi}\int_{0}^{\infty}e^{-z\cosh@@{t}-\nu t}\diff{t}		 | ph ⁡ z | < 1 2 ⁢ π , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence phase 𝑧 1 2 𝜋 𝜈 𝑘 1 0 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi,\Re(\nu+k+1)>% 0}} 
BesselI(nu, z) = (1)/(Pi)*int(exp(z*cos(theta))*cos(nu*theta), theta = 0..Pi)-(sin(nu*Pi))/(Pi)*int(exp(- z*cosh(t)- nu*t), t = 0..infinity)
 
BesselI[\[Nu], z] == Divide[1,Pi]*Integrate[Exp[z*Cos[\[Theta]]]*Cos[\[Nu]*\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None]-Divide[Sin[\[Nu]*Pi],Pi]*Integrate[Exp[- z*Cosh[t]- \[Nu]*t], {t, 0, Infinity}, GenerateConditions->None]
 Failure Aborted Skipped - Because timed out Skipped - Because timed out
10.32.E5 K 0 ⁡ ( z ) = - 1 π ⁢ ∫ 0 π e + z ⁢ cos ⁡ θ ⁢ ( γ + ln ⁡ ( 2 ⁢ z ⁢ ( sin ⁡ θ ) 2 ) ) ⁢ d θ modified-Bessel-second-kind 0 𝑧 1 𝜋 superscript subscript 0 𝜋 superscript 𝑒 𝑧 𝜃 2 𝑧 superscript 𝜃 2 𝜃 {\displaystyle{\displaystyle K_{0}\left(z\right)=-\frac{1}{\pi}\int_{0}^{\pi}e% ^{+z\cos\theta}\left(\gamma+\ln\left(2z(\sin\theta)^{2}\right)\right)\mathrm{d% }\theta}}
\modBesselK{0}@{z} = -\frac{1}{\pi}\int_{0}^{\pi}e^{+ z\cos@@{\theta}}\left(\EulerConstant+\ln@{2z(\sin@@{\theta})^{2}}\right)\diff{\theta}		 

BesselK(0, z) = -(1)/(Pi)*int(exp(+ z*cos(theta))*(gamma + ln(2*z*(sin(theta))^(2))), theta = 0..Pi)
 
BesselK[0, z] == -Divide[1,Pi]*Integrate[Exp[+ z*Cos[\[Theta]]]*(EulerGamma + Log[2*z*(Sin[\[Theta]])^(2)]), {\[Theta], 0, Pi}, GenerateConditions->None]
 Aborted Aborted Skipped - Because timed out Skipped - Because timed out
10.32.E5 K 0 ⁡ ( z ) = - 1 π ⁢ ∫ 0 π e - z ⁢ cos ⁡ θ ⁢ ( γ + ln ⁡ ( 2 ⁢ z ⁢ ( sin ⁡ θ ) 2 ) ) ⁢ d θ modified-Bessel-second-kind 0 𝑧 1 𝜋 superscript subscript 0 𝜋 superscript 𝑒 𝑧 𝜃 2 𝑧 superscript 𝜃 2 𝜃 {\displaystyle{\displaystyle K_{0}\left(z\right)=-\frac{1}{\pi}\int_{0}^{\pi}e% ^{-z\cos\theta}\left(\gamma+\ln\left(2z(\sin\theta)^{2}\right)\right)\mathrm{d% }\theta}}
\modBesselK{0}@{z} = -\frac{1}{\pi}\int_{0}^{\pi}e^{- z\cos@@{\theta}}\left(\EulerConstant+\ln@{2z(\sin@@{\theta})^{2}}\right)\diff{\theta}		 

BesselK(0, z) = -(1)/(Pi)*int(exp(- z*cos(theta))*(gamma + ln(2*z*(sin(theta))^(2))), theta = 0..Pi)
 
BesselK[0, z] == -Divide[1,Pi]*Integrate[Exp[- z*Cos[\[Theta]]]*(EulerGamma + Log[2*z*(Sin[\[Theta]])^(2)]), {\[Theta], 0, Pi}, GenerateConditions->None]
 Aborted Aborted Skipped - Because timed out Skipped - Because timed out
10.32.E6 K 0 ⁡ ( x ) = ∫ 0 ∞ cos ⁡ ( x ⁢ sinh ⁡ t ) ⁢ d t modified-Bessel-second-kind 0 𝑥 superscript subscript 0 𝑥 𝑡 𝑡 {\displaystyle{\displaystyle K_{0}\left(x\right)=\int_{0}^{\infty}\cos\left(x% \sinh t\right)\mathrm{d}t}}
\modBesselK{0}@{x} = \int_{0}^{\infty}\cos@{x\sinh@@{t}}\diff{t}		 x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}} 
BesselK(0, x) = int(cos(x*sinh(t)), t = 0..infinity)
 
BesselK[0, x] == Integrate[Cos[x*Sinh[t]], {t, 0, Infinity}, GenerateConditions->None]
 Successful Aborted - Skipped - Because timed out
10.32.E6 ∫ 0 ∞ cos ⁡ ( x ⁢ sinh ⁡ t ) ⁢ d t = ∫ 0 ∞ cos ⁡ ( x ⁢ t ) t 2 + 1 ⁢ d t superscript subscript 0 𝑥 𝑡 𝑡 superscript subscript 0 𝑥 𝑡 superscript 𝑡 2 1 𝑡 {\displaystyle{\displaystyle\int_{0}^{\infty}\cos\left(x\sinh t\right)\mathrm{% d}t=\int_{0}^{\infty}\frac{\cos\left(xt\right)}{\sqrt{t^{2}+1}}\mathrm{d}t}}
\int_{0}^{\infty}\cos@{x\sinh@@{t}}\diff{t} = \int_{0}^{\infty}\frac{\cos@{xt}}{\sqrt{t^{2}+1}}\diff{t}		 x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}} 
int(cos(x*sinh(t)), t = 0..infinity) = int((cos(x*t))/(sqrt((t)^(2)+ 1)), t = 0..infinity)
 
Integrate[Cos[x*Sinh[t]], {t, 0, Infinity}, GenerateConditions->None] == Integrate[Divide[Cos[x*t],Sqrt[(t)^(2)+ 1]], {t, 0, Infinity}, GenerateConditions->None]
 Successful Aborted - Skipped - Because timed out
10.32.E7 K ν ⁡ ( x ) = sec ⁡ ( 1 2 ⁢ ν ⁢ π ) ⁢ ∫ 0 ∞ cos ⁡ ( x ⁢ sinh ⁡ t ) ⁢ cosh ⁡ ( ν ⁢ t ) ⁢ d t modified-Bessel-second-kind 𝜈 𝑥 1 2 𝜈 𝜋 superscript subscript 0 𝑥 𝑡 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle K_{\nu}\left(x\right)=\sec\left(\tfrac{1}{2}\nu% \pi\right)\int_{0}^{\infty}\cos\left(x\sinh t\right)\cosh\left(\nu t\right)% \mathrm{d}t}}
\modBesselK{\nu}@{x} = \sec@{\tfrac{1}{2}\nu\pi}\int_{0}^{\infty}\cos@{x\sinh@@{t}}\cosh@{\nu t}\diff{t}		 | ℜ ⁡ ν | < 1 , x > 0 formulae-sequence 𝜈 1 𝑥 0 {\displaystyle{\displaystyle|\Re\nu|<1,x>0}} 
BesselK(nu, x) = sec((1)/(2)*nu*Pi)*int(cos(x*sinh(t))*cosh(nu*t), t = 0..infinity)
 
BesselK[\[Nu], x] == Sec[Divide[1,2]*\[Nu]*Pi]*Integrate[Cos[x*Sinh[t]]*Cosh[\[Nu]*t], {t, 0, Infinity}, GenerateConditions->None]
 Successful Aborted Manual Skip! Skipped - Because timed out
10.32.E7 sec ⁡ ( 1 2 ⁢ ν ⁢ π ) ⁢ ∫ 0 ∞ cos ⁡ ( x ⁢ sinh ⁡ t ) ⁢ cosh ⁡ ( ν ⁢ t ) ⁢ d t = csc ⁡ ( 1 2 ⁢ ν ⁢ π ) ⁢ ∫ 0 ∞ sin ⁡ ( x ⁢ sinh ⁡ t ) ⁢ sinh ⁡ ( ν ⁢ t ) ⁢ d t 1 2 𝜈 𝜋 superscript subscript 0 𝑥 𝑡 𝜈 𝑡 𝑡 1 2 𝜈 𝜋 superscript subscript 0 𝑥 𝑡 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle\sec\left(\tfrac{1}{2}\nu\pi\right)\int_{0}^{% \infty}\cos\left(x\sinh t\right)\cosh\left(\nu t\right)\mathrm{d}t=\csc\left(% \tfrac{1}{2}\nu\pi\right)\int_{0}^{\infty}\sin\left(x\sinh t\right)\sinh\left(% \nu t\right)\mathrm{d}t}}
\sec@{\tfrac{1}{2}\nu\pi}\int_{0}^{\infty}\cos@{x\sinh@@{t}}\cosh@{\nu t}\diff{t} = \csc@{\tfrac{1}{2}\nu\pi}\int_{0}^{\infty}\sin@{x\sinh@@{t}}\sinh@{\nu t}\diff{t}		 | ℜ ⁡ ν | < 1 , x > 0 formulae-sequence 𝜈 1 𝑥 0 {\displaystyle{\displaystyle|\Re\nu|<1,x>0}} 
sec((1)/(2)*nu*Pi)*int(cos(x*sinh(t))*cosh(nu*t), t = 0..infinity) = csc((1)/(2)*nu*Pi)*int(sin(x*sinh(t))*sinh(nu*t), t = 0..infinity)
 
Sec[Divide[1,2]*\[Nu]*Pi]*Integrate[Cos[x*Sinh[t]]*Cosh[\[Nu]*t], {t, 0, Infinity}, GenerateConditions->None] == Csc[Divide[1,2]*\[Nu]*Pi]*Integrate[Sin[x*Sinh[t]]*Sinh[\[Nu]*t], {t, 0, Infinity}, GenerateConditions->None]
 Failure Aborted Manual Skip! Skipped - Because timed out
10.32.E8 K ν ⁡ ( z ) = π 1 2 ⁢ ( 1 2 ⁢ z ) ν Γ ⁡ ( ν + 1 2 ) ⁢ ∫ 0 ∞ e - z ⁢ cosh ⁡ t ⁢ ( sinh ⁡ t ) 2 ⁢ ν ⁢ d t modified-Bessel-second-kind 𝜈 𝑧 superscript 𝜋 1 2 superscript 1 2 𝑧 𝜈 Euler-Gamma 𝜈 1 2 superscript subscript 0 superscript 𝑒 𝑧 𝑡 superscript 𝑡 2 𝜈 𝑡 {\displaystyle{\displaystyle K_{\nu}\left(z\right)=\frac{\pi^{\frac{1}{2}}(% \frac{1}{2}z)^{\nu}}{\Gamma\left(\nu+\frac{1}{2}\right)}\int_{0}^{\infty}e^{-z% \cosh t}(\sinh t)^{2\nu}\mathrm{d}t}}
\modBesselK{\nu}@{z} = \frac{\pi^{\frac{1}{2}}(\frac{1}{2}z)^{\nu}}{\EulerGamma@{\nu+\frac{1}{2}}}\int_{0}^{\infty}e^{-z\cosh@@{t}}(\sinh@@{t})^{2\nu}\diff{t}		 ℜ ⁡ ν > - 1 2 , | ph ⁡ z | < 1 2 ⁢ π , ℜ ⁡ ( ν + 1 2 ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence phase 𝑧 1 2 𝜋 𝜈 1 2 0 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},|\operatorname{ph}z|<\tfrac{1% }{2}\pi,\Re(\nu+\frac{1}{2})>0}} 
BesselK(nu, z) = ((Pi)^((1)/(2))*((1)/(2)*z)^(nu))/(GAMMA(nu +(1)/(2)))*int(exp(- z*cosh(t))*(sinh(t))^(2*nu), t = 0..infinity)
 
BesselK[\[Nu], z] == Divide[(Pi)^(Divide[1,2])*(Divide[1,2]*z)^\[Nu],Gamma[\[Nu]+Divide[1,2]]]*Integrate[Exp[- z*Cosh[t]]*(Sinh[t])^(2*\[Nu]), {t, 0, Infinity}, GenerateConditions->None]
 Failure Aborted Skipped - Because timed out Skipped - Because timed out
10.32.E8 π 1 2 ⁢ ( 1 2 ⁢ z ) ν Γ ⁡ ( ν + 1 2 ) ⁢ ∫ 0 ∞ e - z ⁢ cosh ⁡ t ⁢ ( sinh ⁡ t ) 2 ⁢ ν ⁢ d t = π 1 2 ⁢ ( 1 2 ⁢ z ) ν Γ ⁡ ( ν + 1 2 ) ⁢ ∫ 1 ∞ e - z ⁢ t ⁢ ( t 2 - 1 ) ν - 1 2 ⁢ d t superscript 𝜋 1 2 superscript 1 2 𝑧 𝜈 Euler-Gamma 𝜈 1 2 superscript subscript 0 superscript 𝑒 𝑧 𝑡 superscript 𝑡 2 𝜈 𝑡 superscript 𝜋 1 2 superscript 1 2 𝑧 𝜈 Euler-Gamma 𝜈 1 2 superscript subscript 1 superscript 𝑒 𝑧 𝑡 superscript superscript 𝑡 2 1 𝜈 1 2 𝑡 {\displaystyle{\displaystyle\frac{\pi^{\frac{1}{2}}(\frac{1}{2}z)^{\nu}}{% \Gamma\left(\nu+\frac{1}{2}\right)}\int_{0}^{\infty}e^{-z\cosh t}(\sinh t)^{2% \nu}\mathrm{d}t=\frac{\pi^{\frac{1}{2}}(\frac{1}{2}z)^{\nu}}{\Gamma\left(\nu+% \frac{1}{2}\right)}\int_{1}^{\infty}e^{-zt}(t^{2}-1)^{\nu-\frac{1}{2}}\mathrm{% d}t}}
\frac{\pi^{\frac{1}{2}}(\frac{1}{2}z)^{\nu}}{\EulerGamma@{\nu+\frac{1}{2}}}\int_{0}^{\infty}e^{-z\cosh@@{t}}(\sinh@@{t})^{2\nu}\diff{t} = \frac{\pi^{\frac{1}{2}}(\frac{1}{2}z)^{\nu}}{\EulerGamma@{\nu+\frac{1}{2}}}\int_{1}^{\infty}e^{-zt}(t^{2}-1)^{\nu-\frac{1}{2}}\diff{t}		 ℜ ⁡ ν > - 1 2 , | ph ⁡ z | < 1 2 ⁢ π , ℜ ⁡ ( ν + 1 2 ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence phase 𝑧 1 2 𝜋 𝜈 1 2 0 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},|\operatorname{ph}z|<\tfrac{1% }{2}\pi,\Re(\nu+\frac{1}{2})>0}} 
((Pi)^((1)/(2))*((1)/(2)*z)^(nu))/(GAMMA(nu +(1)/(2)))*int(exp(- z*cosh(t))*(sinh(t))^(2*nu), t = 0..infinity) = ((Pi)^((1)/(2))*((1)/(2)*z)^(nu))/(GAMMA(nu +(1)/(2)))*int(exp(- z*t)*((t)^(2)- 1)^(nu -(1)/(2)), t = 1..infinity)
 
Divide[(Pi)^(Divide[1,2])*(Divide[1,2]*z)^\[Nu],Gamma[\[Nu]+Divide[1,2]]]*Integrate[Exp[- z*Cosh[t]]*(Sinh[t])^(2*\[Nu]), {t, 0, Infinity}, GenerateConditions->None] == Divide[(Pi)^(Divide[1,2])*(Divide[1,2]*z)^\[Nu],Gamma[\[Nu]+Divide[1,2]]]*Integrate[Exp[- z*t]*((t)^(2)- 1)^(\[Nu]-Divide[1,2]), {t, 1, Infinity}, GenerateConditions->None]
 Error Aborted Skip - symbolical successful subtest Skipped - Because timed out
10.32.E9 K ν ⁡ ( z ) = ∫ 0 ∞ e - z ⁢ cosh ⁡ t ⁢ cosh ⁡ ( ν ⁢ t ) ⁢ d t modified-Bessel-second-kind 𝜈 𝑧 superscript subscript 0 superscript 𝑒 𝑧 𝑡 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle K_{\nu}\left(z\right)=\int_{0}^{\infty}e^{-z\cosh t% }\cosh\left(\nu t\right)\mathrm{d}t}}
\modBesselK{\nu}@{z} = \int_{0}^{\infty}e^{-z\cosh@@{t}}\cosh@{\nu t}\diff{t}		 | ph ⁡ z | < 1 2 ⁢ π phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi}} 
BesselK(nu, z) = int(exp(- z*cosh(t))*cosh(nu*t), t = 0..infinity)
 
BesselK[\[Nu], z] == Integrate[Exp[- z*Cosh[t]]*Cosh[\[Nu]*t], {t, 0, Infinity}, GenerateConditions->None]
 Failure Aborted Skipped - Because timed out Skipped - Because timed out
10.32.E10 K ν ⁡ ( z ) = 1 2 ⁢ ( 1 2 ⁢ z ) ν ⁢ ∫ 0 ∞ exp ⁡ ( - t - z 2 4 ⁢ t ) ⁢ d t t ν + 1 modified-Bessel-second-kind 𝜈 𝑧 1 2 superscript 1 2 𝑧 𝜈 superscript subscript 0 𝑡 superscript 𝑧 2 4 𝑡 𝑡 superscript 𝑡 𝜈 1 {\displaystyle{\displaystyle K_{\nu}\left(z\right)=\tfrac{1}{2}(\tfrac{1}{2}z)% ^{\nu}\int_{0}^{\infty}\exp\left(-t-\frac{z^{2}}{4t}\right)\frac{\mathrm{d}t}{% t^{\nu+1}}}}
\modBesselK{\nu}@{z} = \tfrac{1}{2}(\tfrac{1}{2}z)^{\nu}\int_{0}^{\infty}\exp@{-t-\frac{z^{2}}{4t}}\frac{\diff{t}}{t^{\nu+1}}		 | ph ⁡ z | < 1 4 ⁢ π phase 𝑧 1 4 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{4}\pi}} 
BesselK(nu, z) = (1)/(2)*((1)/(2)*z)^(nu)* int(exp(- t -((z)^(2))/(4*t))*(1)/((t)^(nu + 1)), t = 0..infinity)
 
BesselK[\[Nu], z] == Divide[1,2]*(Divide[1,2]*z)^\[Nu]* Integrate[Exp[- t -Divide[(z)^(2),4*t]]*Divide[1,(t)^(\[Nu]+ 1)], {t, 0, Infinity}, GenerateConditions->None]
 Successful Successful - Successful [Tested: 40]
10.32.E11 K ν ⁡ ( x ⁢ z ) = Γ ⁡ ( ν + 1 2 ) ⁢ ( 2 ⁢ z ) ν π 1 2 ⁢ x ν ⁢ ∫ 0 ∞ cos ⁡ ( x ⁢ t ) ⁢ d t ( t 2 + z 2 ) ν + 1 2 modified-Bessel-second-kind 𝜈 𝑥 𝑧 Euler-Gamma 𝜈 1 2 superscript 2 𝑧 𝜈 superscript 𝜋 1 2 superscript 𝑥 𝜈 superscript subscript 0 𝑥 𝑡 𝑡 superscript superscript 𝑡 2 superscript 𝑧 2 𝜈 1 2 {\displaystyle{\displaystyle K_{\nu}\left(xz\right)=\frac{\Gamma\left(\nu+% \frac{1}{2}\right)(2z)^{\nu}}{\pi^{\frac{1}{2}}x^{\nu}}\int_{0}^{\infty}\frac{% \cos\left(xt\right)\mathrm{d}t}{(t^{2}+z^{2})^{\nu+\frac{1}{2}}}}}
\modBesselK{\nu}@{xz} = \frac{\EulerGamma@{\nu+\frac{1}{2}}(2z)^{\nu}}{\pi^{\frac{1}{2}}x^{\nu}}\int_{0}^{\infty}\frac{\cos@{xt}\diff{t}}{(t^{2}+z^{2})^{\nu+\frac{1}{2}}}		 ℜ ⁡ ν > - 1 2 , x > 0 , | ph ⁡ z | < 1 2 ⁢ π , ℜ ⁡ ( ν + 1 2 ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence 𝑥 0 formulae-sequence phase 𝑧 1 2 𝜋 𝜈 1 2 0 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},x>0,|\operatorname{ph}z|<% \tfrac{1}{2}\pi,\Re(\nu+\frac{1}{2})>0}} 
BesselK(nu, x*(x + y*I)) = (GAMMA(nu +(1)/(2))*(2*(x + y*I))^(nu))/((Pi)^((1)/(2))* (x)^(nu))*int((cos(x*t))/(((t)^(2)+(x + y*I)^(2))^(nu +(1)/(2))), t = 0..infinity)
 
BesselK[\[Nu], x*(x + y*I)] == Divide[Gamma[\[Nu]+Divide[1,2]]*(2*(x + y*I))^\[Nu],(Pi)^(Divide[1,2])* (x)^\[Nu]]*Integrate[Divide[Cos[x*t],((t)^(2)+(x + y*I)^(2))^(\[Nu]+Divide[1,2])], {t, 0, Infinity}, GenerateConditions->None]
 Error Aborted - Skipped - Because timed out
10.32.E12 I ν ⁡ ( z ) = 1 2 ⁢ π ⁢ i ⁢ ∫ ∞ - i ⁢ π ∞ + i ⁢ π e z ⁢ cosh ⁡ t - ν ⁢ t ⁢ d t modified-Bessel-first-kind 𝜈 𝑧 1 2 𝜋 𝑖 superscript subscript 𝑖 𝜋 𝑖 𝜋 superscript 𝑒 𝑧 𝑡 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle I_{\nu}\left(z\right)=\frac{1}{2\pi i}\int_{% \infty-i\pi}^{\infty+i\pi}e^{z\cosh t-\nu t}\mathrm{d}t}}
\modBesselI{\nu}@{z} = \frac{1}{2\pi i}\int_{\infty-i\pi}^{\infty+i\pi}e^{z\cosh@@{t}-\nu t}\diff{t}		 | ph ⁡ z | < 1 2 ⁢ π , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence phase 𝑧 1 2 𝜋 𝜈 𝑘 1 0 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi,\Re(\nu+k+1)>% 0}} 
BesselI(nu, z) = (1)/(2*Pi*I)*int(exp(z*cosh(t)- nu*t), t = infinity - I*Pi..infinity + I*Pi)
 
BesselI[\[Nu], z] == Divide[1,2*Pi*I]*Integrate[Exp[z*Cosh[t]- \[Nu]*t], {t, Infinity - I*Pi, Infinity + I*Pi}, GenerateConditions->None]
 Error Failure -
Failed [50 / 50]
Result: Complex[0.5303418993681409, 0.010453999760907294]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[1.7664848208906112, 0.1468422559210476]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.32.E13 K ν ⁡ ( z ) = ( 1 2 ⁢ z ) ν 4 ⁢ π ⁢ i ⁢ ∫ c - i ⁢ ∞ c + i ⁢ ∞ Γ ⁡ ( t ) ⁢ Γ ⁡ ( t - ν ) ⁢ ( 1 2 ⁢ z ) - 2 ⁢ t ⁢ d t modified-Bessel-second-kind 𝜈 𝑧 superscript 1 2 𝑧 𝜈 4 𝜋 𝑖 superscript subscript 𝑐 𝑖 𝑐 𝑖 Euler-Gamma 𝑡 Euler-Gamma 𝑡 𝜈 superscript 1 2 𝑧 2 𝑡 𝑡 {\displaystyle{\displaystyle K_{\nu}\left(z\right)=\frac{(\frac{1}{2}z)^{\nu}}% {4\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma\left(t\right)\Gamma\left(t-\nu% \right)(\tfrac{1}{2}z)^{-2t}\mathrm{d}t}}
\modBesselK{\nu}@{z} = \frac{(\frac{1}{2}z)^{\nu}}{4\pi i}\int_{c-i\infty}^{c+i\infty}\EulerGamma@{t}\EulerGamma@{t-\nu}(\tfrac{1}{2}z)^{-2t}\diff{t}		 c > max ⁡ ( ℜ ⁡ ν , 0 ) < 1 2 ⁢ π , | ph ⁡ z | < 1 2 ⁢ π , ℜ ⁡ t > 0 , ℜ ⁡ ( t - ν ) > 0 formulae-sequence 𝑐 𝜈 0 1 2 𝜋 formulae-sequence phase 𝑧 1 2 𝜋 formulae-sequence 𝑡 0 𝑡 𝜈 0 {\displaystyle{\displaystyle c>\max(\Re\nu,0)<\frac{1}{2}\pi,|\operatorname{ph% }z|<\frac{1}{2}\pi,\Re t>0,\Re(t-\nu)>0}} 
BesselK(nu, z) = (((1)/(2)*z)^(nu))/(4*Pi*I)*int(GAMMA(t)*GAMMA(t - nu)*((1)/(2)*z)^(- 2*t), t = c - I*infinity..c + I*infinity)
 
BesselK[\[Nu], z] == Divide[(Divide[1,2]*z)^\[Nu],4*Pi*I]*Integrate[Gamma[t]*Gamma[t - \[Nu]]*(Divide[1,2]*z)^(- 2*t), {t, c - I*Infinity, c + I*Infinity}, GenerateConditions->None]
 Failure Aborted
Failed [300 / 300]
Result: .5663982443-.3181066824*I
Test Values: {c = -3/2, nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}

Result: -1.434992817-2.759712160*I
Test Values: {c = -3/2, nu = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Skipped - Because timed out
10.32.E14 K ν ⁡ ( z ) = 1 2 ⁢ π 2 ⁢ i ⁢ ( π 2 ⁢ z ) 1 2 ⁢ e - z ⁢ cos ⁡ ( ν ⁢ π ) ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ Γ ⁡ ( t ) ⁢ Γ ⁡ ( 1 2 - t - ν ) ⁢ Γ ⁡ ( 1 2 - t + ν ) ⁢ ( 2 ⁢ z ) t ⁢ d t modified-Bessel-second-kind 𝜈 𝑧 1 2 superscript 𝜋 2 𝑖 superscript 𝜋 2 𝑧 1 2 superscript 𝑒 𝑧 𝜈 𝜋 superscript subscript 𝑖 𝑖 Euler-Gamma 𝑡 Euler-Gamma 1 2 𝑡 𝜈 Euler-Gamma 1 2 𝑡 𝜈 superscript 2 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle K_{\nu}\left(z\right)=\frac{1}{2\pi^{2}i}\left(% \frac{\pi}{2z}\right)^{\frac{1}{2}}e^{-z}\cos\left(\nu\pi\right)\*\int_{-i% \infty}^{i\infty}\Gamma\left(t\right)\Gamma\left(\tfrac{1}{2}-t-\nu\right)% \Gamma\left(\tfrac{1}{2}-t+\nu\right)(2z)^{t}\mathrm{d}t}}
\modBesselK{\nu}@{z} = \frac{1}{2\pi^{2}i}\left(\frac{\pi}{2z}\right)^{\frac{1}{2}}e^{-z}\cos@{\nu\pi}\*\int_{-i\infty}^{i\infty}\EulerGamma@{t}\EulerGamma@{\tfrac{1}{2}-t-\nu}\EulerGamma@{\tfrac{1}{2}-t+\nu}(2z)^{t}\diff{t}		 ν - 1 2 ∉ ℤ < 3 2 ⁢ π , | ph ⁡ z | < 3 2 ⁢ π , ℜ ⁡ t > 0 , ℜ ⁡ ( 1 2 - t - ν ) > 0 , ℜ ⁡ ( 1 2 - t + ν ) > 0 formulae-sequence 𝜈 1 2 3 2 𝜋 formulae-sequence phase 𝑧 3 2 𝜋 formulae-sequence 𝑡 0 formulae-sequence 1 2 𝑡 𝜈 0 1 2 𝑡 𝜈 0 {\displaystyle{\displaystyle\nu-\tfrac{1}{2}\notin\mathbb{Z}<\tfrac{3}{2}\pi,|% \operatorname{ph}z|<\tfrac{3}{2}\pi,\Re t>0,\Re(\tfrac{1}{2}-t-\nu)>0,\Re(% \tfrac{1}{2}-t+\nu)>0}} 
BesselK(nu, z) = (1)/(2*(Pi)^(2)* I)*((Pi)/(2*z))^((1)/(2))* exp(- z)*cos(nu*Pi)* int(GAMMA(t)*GAMMA((1)/(2)- t - nu)*GAMMA((1)/(2)- t + nu)*(2*z)^(t), t = - I*infinity..I*infinity)
 
BesselK[\[Nu], z] == Divide[1,2*(Pi)^(2)* I]*(Divide[Pi,2*z])^(Divide[1,2])* Exp[- z]*Cos[\[Nu]*Pi]* Integrate[Gamma[t]*Gamma[Divide[1,2]- t - \[Nu]]*Gamma[Divide[1,2]- t + \[Nu]]*(2*z)^(t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]
 Failure Aborted Skipped - Because timed out Skipped - Because timed out
10.32.E15 I μ ⁡ ( z ) ⁢ I ν ⁡ ( z ) = 2 π ⁢ ∫ 0 1 2 ⁢ π I μ + ν ⁡ ( 2 ⁢ z ⁢ cos ⁡ θ ) ⁢ cos ⁡ ( ( μ - ν ) ⁢ θ ) ⁢ d θ modified-Bessel-first-kind 𝜇 𝑧 modified-Bessel-first-kind 𝜈 𝑧 2 𝜋 superscript subscript 0 1 2 𝜋 modified-Bessel-first-kind 𝜇 𝜈 2 𝑧 𝜃 𝜇 𝜈 𝜃 𝜃 {\displaystyle{\displaystyle I_{\mu}\left(z\right)I_{\nu}\left(z\right)=\frac{% 2}{\pi}\int_{0}^{\frac{1}{2}\pi}I_{\mu+\nu}\left(2z\cos\theta\right)\cos\left(% (\mu-\nu)\theta\right)\mathrm{d}\theta}}
\modBesselI{\mu}@{z}\modBesselI{\nu}@{z} = \frac{2}{\pi}\int_{0}^{\frac{1}{2}\pi}\modBesselI{\mu+\nu}@{2z\cos@@{\theta}}\cos@{(\mu-\nu)\theta}\diff{\theta}		 ℜ ⁡ ( μ + ν ) > - 1 , ℜ ⁡ ( ( μ ) + k + 1 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( μ + ν ) + k + 1 ) > 0 formulae-sequence 𝜇 𝜈 1 formulae-sequence 𝜇 𝑘 1 0 formulae-sequence 𝜈 𝑘 1 0 𝜇 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\left(\mu+\nu\right)>-1,\Re((\mu)+k+1)>0,\Re(% \nu+k+1)>0,\Re((\mu+\nu)+k+1)>0}} 
BesselI(mu, z)*BesselI(nu, z) = (2)/(Pi)*int(BesselI(mu + nu, 2*z*cos(theta))*cos((mu - nu)*theta), theta = 0..(1)/(2)*Pi)
 
BesselI[\[Mu], z]*BesselI[\[Nu], z] == Divide[2,Pi]*Integrate[BesselI[\[Mu]+ \[Nu], 2*z*Cos[\[Theta]]]*Cos[(\[Mu]- \[Nu])*\[Theta]], {\[Theta], 0, Divide[1,2]*Pi}, GenerateConditions->None]
 Failure Aborted Skipped - Because timed out Skipped - Because timed out
10.32.E16 I μ ⁡ ( x ) ⁢ K ν ⁡ ( x ) = ∫ 0 ∞ J μ + ν ⁡ ( 2 ⁢ x ⁢ sinh ⁡ t ) ⁢ e ( - μ + ν ) ⁢ t ⁢ d t modified-Bessel-first-kind 𝜇 𝑥 modified-Bessel-second-kind 𝜈 𝑥 superscript subscript 0 Bessel-J 𝜇 𝜈 2 𝑥 𝑡 superscript 𝑒 𝜇 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle I_{\mu}\left(x\right)K_{\nu}\left(x\right)=\int_{% 0}^{\infty}J_{\mu+\nu}\left(2x\sinh t\right)e^{(-\mu+\nu)t}\mathrm{d}t}}
\modBesselI{\mu}@{x}\modBesselK{\nu}@{x} = \int_{0}^{\infty}\BesselJ{\mu+\nu}@{2x\sinh@@{t}}e^{(-\mu+\nu)t}\diff{t}		 ℜ ⁡ ( μ - ν ) > - 1 2 , ℜ ⁡ ( μ + ν ) > - 1 2 , ℜ ⁡ ( μ + ν ) > - 1 , ℜ ⁡ ( μ - ν ) > - 1 , x > 0 , ℜ ⁡ ( ( μ + ν ) + k + 1 ) > 0 , ℜ ⁡ ( ( μ ) + k + 1 ) > 0 formulae-sequence 𝜇 𝜈 1 2 formulae-sequence 𝜇 𝜈 1 2 formulae-sequence 𝜇 𝜈 1 formulae-sequence 𝜇 𝜈 1 formulae-sequence 𝑥 0 formulae-sequence 𝜇 𝜈 𝑘 1 0 𝜇 𝑘 1 0 {\displaystyle{\displaystyle\Re\left(\mu-\nu\right)>-\tfrac{1}{2},\Re\left(\mu% +\nu\right)>-\tfrac{1}{2},\Re\left(\mu+\nu\right)>-1,\Re\left(\mu-\nu\right)>-% 1,x>0,\Re((\mu+\nu)+k+1)>0,\Re((\mu)+k+1)>0}} 
BesselI(mu, x)*BesselK(nu, x) = int(BesselJ(mu + nu, 2*x*sinh(t))*exp((- mu + nu)*t), t = 0..infinity)
 
BesselI[\[Mu], x]*BesselK[\[Nu], x] == Integrate[BesselJ[\[Mu]+ \[Nu], 2*x*Sinh[t]]*Exp[(- \[Mu]+ \[Nu])*t], {t, 0, Infinity}, GenerateConditions->None]
 Error Aborted - Skipped - Because timed out
10.32.E16 I μ ⁡ ( x ) ⁢ K ν ⁡ ( x ) = ∫ 0 ∞ J μ - ν ⁡ ( 2 ⁢ x ⁢ sinh ⁡ t ) ⁢ e ( - μ - ν ) ⁢ t ⁢ d t modified-Bessel-first-kind 𝜇 𝑥 modified-Bessel-second-kind 𝜈 𝑥 superscript subscript 0 Bessel-J 𝜇 𝜈 2 𝑥 𝑡 superscript 𝑒 𝜇 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle I_{\mu}\left(x\right)K_{\nu}\left(x\right)=\int_{% 0}^{\infty}J_{\mu-\nu}\left(2x\sinh t\right)e^{(-\mu-\nu)t}\mathrm{d}t}}
\modBesselI{\mu}@{x}\modBesselK{\nu}@{x} = \int_{0}^{\infty}\BesselJ{\mu-\nu}@{2x\sinh@@{t}}e^{(-\mu-\nu)t}\diff{t}		 ℜ ⁡ ( μ - ν ) > - 1 2 , ℜ ⁡ ( μ + ν ) > - 1 2 , ℜ ⁡ ( μ + ν ) > - 1 , ℜ ⁡ ( μ - ν ) > - 1 , x > 0 , ℜ ⁡ ( ( μ + ν ) + k + 1 ) > 0 , ℜ ⁡ ( ( μ ) + k + 1 ) > 0 , ℜ ⁡ ( ( μ - ν ) + k + 1 ) > 0 formulae-sequence 𝜇 𝜈 1 2 formulae-sequence 𝜇 𝜈 1 2 formulae-sequence 𝜇 𝜈 1 formulae-sequence 𝜇 𝜈 1 formulae-sequence 𝑥 0 formulae-sequence 𝜇 𝜈 𝑘 1 0 formulae-sequence 𝜇 𝑘 1 0 𝜇 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\left(\mu-\nu\right)>-\tfrac{1}{2},\Re\left(\mu% +\nu\right)>-\tfrac{1}{2},\Re\left(\mu+\nu\right)>-1,\Re\left(\mu-\nu\right)>-% 1,x>0,\Re((\mu+\nu)+k+1)>0,\Re((\mu)+k+1)>0,\Re((\mu-\nu)+k+1)>0}} 
BesselI(mu, x)*BesselK(nu, x) = int(BesselJ(mu - nu, 2*x*sinh(t))*exp((- mu - nu)*t), t = 0..infinity)
 
BesselI[\[Mu], x]*BesselK[\[Nu], x] == Integrate[BesselJ[\[Mu]- \[Nu], 2*x*Sinh[t]]*Exp[(- \[Mu]- \[Nu])*t], {t, 0, Infinity}, GenerateConditions->None]
 Error Aborted - Skipped - Because timed out
10.32.E17 K μ ⁡ ( z ) ⁢ K ν ⁡ ( z ) = 2 ⁢ ∫ 0 ∞ K μ + ν ⁡ ( 2 ⁢ z ⁢ cosh ⁡ t ) ⁢ cosh ⁡ ( ( μ - ν ) ⁢ t ) ⁢ d t modified-Bessel-second-kind 𝜇 𝑧 modified-Bessel-second-kind 𝜈 𝑧 2 superscript subscript 0 modified-Bessel-second-kind 𝜇 𝜈 2 𝑧 𝑡 𝜇 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle K_{\mu}\left(z\right)K_{\nu}\left(z\right)=2\int_% {0}^{\infty}K_{\mu+\nu}\left(2z\cosh t\right)\cosh\left((\mu-\nu)t\right)% \mathrm{d}t}}
\modBesselK{\mu}@{z}\modBesselK{\nu}@{z} = 2\int_{0}^{\infty}\modBesselK{\mu+\nu}@{2z\cosh@@{t}}\cosh@{(\mu-\nu)t}\diff{t}		 | ph ⁡ z | < 1 2 ⁢ π phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi}} 
BesselK(mu, z)*BesselK(nu, z) = 2*int(BesselK(mu + nu, 2*z*cosh(t))*cosh((mu - nu)*t), t = 0..infinity)
 
BesselK[\[Mu], z]*BesselK[\[Nu], z] == 2*Integrate[BesselK[\[Mu]+ \[Nu], 2*z*Cosh[t]]*Cosh[(\[Mu]- \[Nu])*t], {t, 0, Infinity}, GenerateConditions->None]
 Failure Aborted Manual Skip! Skipped - Because timed out
10.32.E17 K μ ⁡ ( z ) ⁢ K ν ⁡ ( z ) = 2 ⁢ ∫ 0 ∞ K μ - ν ⁡ ( 2 ⁢ z ⁢ cosh ⁡ t ) ⁢ cosh ⁡ ( ( μ + ν ) ⁢ t ) ⁢ d t modified-Bessel-second-kind 𝜇 𝑧 modified-Bessel-second-kind 𝜈 𝑧 2 superscript subscript 0 modified-Bessel-second-kind 𝜇 𝜈 2 𝑧 𝑡 𝜇 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle K_{\mu}\left(z\right)K_{\nu}\left(z\right)=2\int_% {0}^{\infty}K_{\mu-\nu}\left(2z\cosh t\right)\cosh\left((\mu+\nu)t\right)% \mathrm{d}t}}
\modBesselK{\mu}@{z}\modBesselK{\nu}@{z} = 2\int_{0}^{\infty}\modBesselK{\mu-\nu}@{2z\cosh@@{t}}\cosh@{(\mu+\nu)t}\diff{t}		 | ph ⁡ z | < 1 2 ⁢ π phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi}} 
BesselK(mu, z)*BesselK(nu, z) = 2*int(BesselK(mu - nu, 2*z*cosh(t))*cosh((mu + nu)*t), t = 0..infinity)
 
BesselK[\[Mu], z]*BesselK[\[Nu], z] == 2*Integrate[BesselK[\[Mu]- \[Nu], 2*z*Cosh[t]]*Cosh[(\[Mu]+ \[Nu])*t], {t, 0, Infinity}, GenerateConditions->None]
 Failure Aborted Manual Skip! Skipped - Because timed out
10.32.E18 K ν ⁡ ( z ) ⁢ K ν ⁡ ( ζ ) = 1 2 ⁢ ∫ 0 ∞ exp ⁡ ( - t 2 - z 2 + ζ 2 2 ⁢ t ) ⁢ K ν ⁡ ( z ⁢ ζ t ) ⁢ d t t modified-Bessel-second-kind 𝜈 𝑧 modified-Bessel-second-kind 𝜈 𝜁 1 2 superscript subscript 0 𝑡 2 superscript 𝑧 2 superscript 𝜁 2 2 𝑡 modified-Bessel-second-kind 𝜈 𝑧 𝜁 𝑡 𝑡 𝑡 {\displaystyle{\displaystyle K_{\nu}\left(z\right)K_{\nu}\left(\zeta\right)=% \frac{1}{2}\int_{0}^{\infty}\exp\left(-\frac{t}{2}-\frac{z^{2}+\zeta^{2}}{2t}% \right)K_{\nu}\left(\frac{z\zeta}{t}\right)\frac{\mathrm{d}t}{t}}}
\modBesselK{\nu}@{z}\modBesselK{\nu}@{\zeta} = \frac{1}{2}\int_{0}^{\infty}\exp@{-\frac{t}{2}-\frac{z^{2}+\zeta^{2}}{2t}}\modBesselK{\nu}\left(\frac{z\zeta}{t}\right)\frac{\diff{t}}{t}		 | ph ⁡ z | < π , | ph ⁡ ζ | < π , | ph ⁡ ( z + ζ ) | < 1 4 ⁢ π formulae-sequence phase 𝑧 𝜋 formulae-sequence phase 𝜁 𝜋 phase 𝑧 𝜁 1 4 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\pi,|\operatorname{ph}\zeta|<% \pi,|\operatorname{ph}\left(z+\zeta\right)|<\tfrac{1}{4}\pi}} 
BesselK(nu, z)*BesselK(nu, zeta) = (1)/(2)*int(exp(-(t)/(2)-((z)^(2)+ (zeta)^(2))/(2*t))*BesselK(nu, (z*zeta)/(t))*(1)/(t), t = 0..infinity)
 
BesselK[\[Nu], z]*BesselK[\[Nu], \[Zeta]] == Divide[1,2]*Integrate[Exp[-Divide[t,2]-Divide[(z)^(2)+ \[Zeta]^(2),2*t]]*BesselK[\[Nu], Divide[z*\[Zeta],t]]*Divide[1,t], {t, 0, Infinity}, GenerateConditions->None]
 Translation Error Translation Error - -
10.32.E19 K μ ⁡ ( z ) ⁢ K ν ⁡ ( z ) = 1 8 ⁢ π ⁢ i ⁢ ∫ c - i ⁢ ∞ c + i ⁢ ∞ Γ ⁡ ( t + 1 2 ⁢ μ + 1 2 ⁢ ν ) ⁢ Γ ⁡ ( t + 1 2 ⁢ μ - 1 2 ⁢ ν ) ⁢ Γ ⁡ ( t - 1 2 ⁢ μ + 1 2 ⁢ ν ) ⁢ Γ ⁡ ( t - 1 2 ⁢ μ - 1 2 ⁢ ν ) Γ ⁡ ( 2 ⁢ t ) ⁢ ( 1 2 ⁢ z ) - 2 ⁢ t ⁢ d t modified-Bessel-second-kind 𝜇 𝑧 modified-Bessel-second-kind 𝜈 𝑧 1 8 𝜋 𝑖 superscript subscript 𝑐 𝑖 𝑐 𝑖 Euler-Gamma 𝑡 1 2 𝜇 1 2 𝜈 Euler-Gamma 𝑡 1 2 𝜇 1 2 𝜈 Euler-Gamma 𝑡 1 2 𝜇 1 2 𝜈 Euler-Gamma 𝑡 1 2 𝜇 1 2 𝜈 Euler-Gamma 2 𝑡 superscript 1 2 𝑧 2 𝑡 𝑡 {\displaystyle{\displaystyle K_{\mu}\left(z\right)K_{\nu}\left(z\right)=\frac{% 1}{8\pi i}\int_{c-i\infty}^{c+i\infty}\frac{\Gamma\left(t+\frac{1}{2}\mu+\frac% {1}{2}\nu\right)\Gamma\left(t+\frac{1}{2}\mu-\frac{1}{2}\nu\right)\Gamma\left(% t-\frac{1}{2}\mu+\frac{1}{2}\nu\right)\Gamma\left(t-\frac{1}{2}\mu-\frac{1}{2}% \nu\right)}{\Gamma\left(2t\right)}(\tfrac{1}{2}z)^{-2t}\mathrm{d}t}}
\modBesselK{\mu}@{z}\modBesselK{\nu}@{z} = \frac{1}{8\pi i}\int_{c-i\infty}^{c+i\infty}\frac{\EulerGamma@{t+\frac{1}{2}\mu+\frac{1}{2}\nu}\EulerGamma@{t+\frac{1}{2}\mu-\frac{1}{2}\nu}\EulerGamma@{t-\frac{1}{2}\mu+\frac{1}{2}\nu}\EulerGamma@{t-\frac{1}{2}\mu-\frac{1}{2}\nu}}{\EulerGamma@{2t}}(\tfrac{1}{2}z)^{-2t}\diff{t}		 c > 1 2 ⁢ ( | ℜ ⁡ μ | + | ℜ ⁡ ν | ) , | ph ⁡ z | < 1 2 ⁢ π , ℜ ⁡ ( t + 1 2 ⁢ μ + 1 2 ⁢ ν ) > 0 , ℜ ⁡ ( t + 1 2 ⁢ μ - 1 2 ⁢ ν ) > 0 , ℜ ⁡ ( t - 1 2 ⁢ μ + 1 2 ⁢ ν ) > 0 , ℜ ⁡ ( t - 1 2 ⁢ μ - 1 2 ⁢ ν ) > 0 , ℜ ⁡ ( 2 ⁢ t ) > 0 formulae-sequence 𝑐 1 2 𝜇 𝜈 formulae-sequence phase 𝑧 1 2 𝜋 formulae-sequence 𝑡 1 2 𝜇 1 2 𝜈 0 formulae-sequence 𝑡 1 2 𝜇 1 2 𝜈 0 formulae-sequence 𝑡 1 2 𝜇 1 2 𝜈 0 formulae-sequence 𝑡 1 2 𝜇 1 2 𝜈 0 2 𝑡 0 {\displaystyle{\displaystyle c>\tfrac{1}{2}(|\Re\mu|+|\Re\nu|),|\operatorname{% ph}z|<\tfrac{1}{2}\pi,\Re(t+\frac{1}{2}\mu+\frac{1}{2}\nu)>0,\Re(t+\frac{1}{2}% \mu-\frac{1}{2}\nu)>0,\Re(t-\frac{1}{2}\mu+\frac{1}{2}\nu)>0,\Re(t-\frac{1}{2}% \mu-\frac{1}{2}\nu)>0,\Re(2t)>0}} 
BesselK(mu, z)*BesselK(nu, z) = (1)/(8*Pi*I)*int((GAMMA(t +(1)/(2)*mu +(1)/(2)*nu)*GAMMA(t +(1)/(2)*mu -(1)/(2)*nu)*GAMMA(t -(1)/(2)*mu +(1)/(2)*nu)*GAMMA(t -(1)/(2)*mu -(1)/(2)*nu))/(GAMMA(2*t))*((1)/(2)*z)^(- 2*t), t = c - I*infinity..c + I*infinity)
 
BesselK[\[Mu], z]*BesselK[\[Nu], z] == Divide[1,8*Pi*I]*Integrate[Divide[Gamma[t +Divide[1,2]*\[Mu]+Divide[1,2]*\[Nu]]*Gamma[t +Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]]*Gamma[t -Divide[1,2]*\[Mu]+Divide[1,2]*\[Nu]]*Gamma[t -Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]],Gamma[2*t]]*(Divide[1,2]*z)^(- 2*t), {t, c - I*Infinity, c + I*Infinity}, GenerateConditions->None]
 Error Aborted - Skip - No test values generated
10.34.E1 I ν ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = e m ⁢ ν ⁢ π ⁢ i ⁢ I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 superscript 𝑒 𝑚 𝜈 𝜋 𝑖 modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\nu}\left(ze^{m\pi i}\right)=e^{m\nu\pi i}I_{% \nu}\left(z\right)}}
\modBesselI{\nu}@{ze^{m\pi i}} = e^{m\nu\pi i}\modBesselI{\nu}@{z}		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
BesselI(nu, z*exp(m*Pi*I)) = exp(m*nu*Pi*I)*BesselI(nu, z)
 
BesselI[\[Nu], z*Exp[m*Pi*I]] == Exp[m*\[Nu]*Pi*I]*BesselI[\[Nu], z]
 Failure Failure
Failed [132 / 210]
Result: -2.206479866-1.131319388*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 1}

Result: .5147384726+.2724622562e-1*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 2}

... skip entries to safe data
Failed [120 / 210]
Result: Complex[-2.206479866313521, -1.1313193889480602]
Test Values: {Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[0.5147384728800724, 0.02724622519878004]
Test Values: {Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.34.E2 K ν ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = e - m ⁢ ν ⁢ π ⁢ i ⁢ K ν ⁡ ( z ) - π ⁢ i ⁢ sin ⁡ ( m ⁢ ν ⁢ π ) ⁢ csc ⁡ ( ν ⁢ π ) ⁢ I ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 superscript 𝑒 𝑚 𝜈 𝜋 𝑖 modified-Bessel-second-kind 𝜈 𝑧 𝜋 𝑖 𝑚 𝜈 𝜋 𝜈 𝜋 modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\nu}\left(ze^{m\pi i}\right)=e^{-m\nu\pi i}K_{% \nu}\left(z\right)-\pi i\sin\left(m\nu\pi\right)\csc\left(\nu\pi\right)I_{\nu}% \left(z\right)}}
\modBesselK{\nu}@{ze^{m\pi i}} = e^{-m\nu\pi i}\modBesselK{\nu}@{z}-\pi i\sin@{m\nu\pi}\csc@{\nu\pi}\modBesselI{\nu}@{z}		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
BesselK(nu, z*exp(m*Pi*I)) = exp(- m*nu*Pi*I)*BesselK(nu, z)- Pi*I*sin(m*nu*Pi)*csc(nu*Pi)*BesselI(nu, z)
 
BesselK[\[Nu], z*Exp[m*Pi*I]] == Exp[- m*\[Nu]*Pi*I]*BesselK[\[Nu], z]- Pi*I*Sin[m*\[Nu]*Pi]*Csc[\[Nu]*Pi]*BesselI[\[Nu], z]
 Failure Failure
Failed [170 / 210]
Result: 2.965939338+3.157233720*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 1}

Result: -10.37113928-12.75980866*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 2}

... skip entries to safe data
Failed [162 / 210]
Result: Complex[2.965939340334436, 3.157233721966529]
Test Values: {Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-10.371139260352992, -12.75980869099896]
Test Values: {Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.34.E3 I ν ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = ( i / π ) ⁢ ( + e m ⁢ ν ⁢ π ⁢ i ⁢ K ν ⁡ ( z ⁢ e + π ⁢ i ) - e ( m - 1 ) ⁢ ν ⁢ π ⁢ i ⁢ K ν ⁡ ( z ) ) modified-Bessel-first-kind 𝜈 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 𝑖 𝜋 superscript 𝑒 𝑚 𝜈 𝜋 𝑖 modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 superscript 𝑒 𝑚 1 𝜈 𝜋 𝑖 modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\nu}\left(ze^{m\pi i}\right)=(i/\pi)\left(+e^{% m\nu\pi i}K_{\nu}\left(ze^{+\pi i}\right)-e^{(m-1)\nu\pi i}K_{\nu}\left(z% \right)\right)}}
\modBesselI{\nu}@{ze^{m\pi i}} = (i/\pi)\left(+ e^{m\nu\pi i}\modBesselK{\nu}@{ze^{+\pi i}}- e^{(m- 1)\nu\pi i}\modBesselK{\nu}@{z}\right)		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
BesselI(nu, z*exp(m*Pi*I)) = (I/Pi)*(+ exp(m*nu*Pi*I)*BesselK(nu, z*exp(+ Pi*I))- exp((m - 1)*nu*Pi*I)*BesselK(nu, z))
 
BesselI[\[Nu], z*Exp[m*Pi*I]] == (I/Pi)*(+ Exp[m*\[Nu]*Pi*I]*BesselK[\[Nu], z*Exp[+ Pi*I]]- Exp[(m - 1)*\[Nu]*Pi*I]*BesselK[\[Nu], z])
 Failure Failure
Failed [152 / 210]
Result: -2.316975457-.8668337446*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 1}

Result: .5132395470-.3232131754e-1*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 2}

... skip entries to safe data
Failed [140 / 210]
Result: Complex[-2.3169754573845194, -0.8668337451474188]
Test Values: {Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[0.5132395471581521, -0.03232131806579792]
Test Values: {Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.34.E3 I ν ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = ( i / π ) ⁢ ( - e m ⁢ ν ⁢ π ⁢ i ⁢ K ν ⁡ ( z ⁢ e - π ⁢ i ) + e ( m + 1 ) ⁢ ν ⁢ π ⁢ i ⁢ K ν ⁡ ( z ) ) modified-Bessel-first-kind 𝜈 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 𝑖 𝜋 superscript 𝑒 𝑚 𝜈 𝜋 𝑖 modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 superscript 𝑒 𝑚 1 𝜈 𝜋 𝑖 modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\nu}\left(ze^{m\pi i}\right)=(i/\pi)\left(-e^{% m\nu\pi i}K_{\nu}\left(ze^{-\pi i}\right)+e^{(m+1)\nu\pi i}K_{\nu}\left(z% \right)\right)}}
\modBesselI{\nu}@{ze^{m\pi i}} = (i/\pi)\left(- e^{m\nu\pi i}\modBesselK{\nu}@{ze^{-\pi i}}+ e^{(m+ 1)\nu\pi i}\modBesselK{\nu}@{z}\right)		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
BesselI(nu, z*exp(m*Pi*I)) = (I/Pi)*(- exp(m*nu*Pi*I)*BesselK(nu, z*exp(- Pi*I))+ exp((m + 1)*nu*Pi*I)*BesselK(nu, z))
 
BesselI[\[Nu], z*Exp[m*Pi*I]] == (I/Pi)*(- Exp[m*\[Nu]*Pi*I]*BesselK[\[Nu], z*Exp[- Pi*I]]+ Exp[(m + 1)*\[Nu]*Pi*I]*BesselK[\[Nu], z])
 Failure Failure
Failed [190 / 210]
Result: -2.206479866-1.131319388*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 1}

Result: .5147384726+.2724622561e-1*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 2}

... skip entries to safe data
Failed [190 / 210]
Result: Complex[-2.206479866313521, -1.1313193889480602]
Test Values: {Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[0.5147384728800724, 0.027246225198780036]
Test Values: {Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.34.E4 K ν ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = csc ⁡ ( ν ⁢ π ) ⁢ ( + sin ⁡ ( m ⁢ ν ⁢ π ) ⁢ K ν ⁡ ( z ⁢ e + π ⁢ i ) - sin ⁡ ( ( m - 1 ) ⁢ ν ⁢ π ) ⁢ K ν ⁡ ( z ) ) modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 𝜈 𝜋 𝑚 𝜈 𝜋 modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 𝑚 1 𝜈 𝜋 modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\nu}\left(ze^{m\pi i}\right)=\csc\left(\nu\pi% \right)\left(+\sin\left(m\nu\pi\right)K_{\nu}\left(ze^{+\pi i}\right)-\sin% \left((m-1)\nu\pi\right)K_{\nu}\left(z\right)\right)}}
\modBesselK{\nu}@{ze^{m\pi i}} = \csc@{\nu\pi}\left(+\sin@{m\nu\pi}\modBesselK{\nu}@{ze^{+\pi i}}-\sin@{(m- 1)\nu\pi}\modBesselK{\nu}@{z}\right)		 

BesselK(nu, z*exp(m*Pi*I)) = csc(nu*Pi)*(+ sin(m*nu*Pi)*BesselK(nu, z*exp(+ Pi*I))- sin((m - 1)*nu*Pi)*BesselK(nu, z))
 
BesselK[\[Nu], z*Exp[m*Pi*I]] == Csc[\[Nu]*Pi]*(+ Sin[m*\[Nu]*Pi]*BesselK[\[Nu], z*Exp[+ Pi*I]]- Sin[(m - 1)*\[Nu]*Pi]*BesselK[\[Nu], z])
 Failure Failure
Failed [158 / 210]
Result: -2.723238516+7.278993081*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 2}

Result: 29.12762958-25.06220737*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 3}

... skip entries to safe data
Failed [154 / 210]
Result: Complex[-2.7232385256388585, 7.278993075467058]
Test Values: {Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[29.127629620508102, -25.062207299552764]
Test Values: {Rule[m, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.34.E4 K ν ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = csc ⁡ ( ν ⁢ π ) ⁢ ( - sin ⁡ ( m ⁢ ν ⁢ π ) ⁢ K ν ⁡ ( z ⁢ e - π ⁢ i ) + sin ⁡ ( ( m + 1 ) ⁢ ν ⁢ π ) ⁢ K ν ⁡ ( z ) ) modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 𝜈 𝜋 𝑚 𝜈 𝜋 modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 𝑚 1 𝜈 𝜋 modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\nu}\left(ze^{m\pi i}\right)=\csc\left(\nu\pi% \right)\left(-\sin\left(m\nu\pi\right)K_{\nu}\left(ze^{-\pi i}\right)+\sin% \left((m+1)\nu\pi\right)K_{\nu}\left(z\right)\right)}}
\modBesselK{\nu}@{ze^{m\pi i}} = \csc@{\nu\pi}\left(-\sin@{m\nu\pi}\modBesselK{\nu}@{ze^{-\pi i}}+\sin@{(m+ 1)\nu\pi}\modBesselK{\nu}@{z}\right)		 

BesselK(nu, z*exp(m*Pi*I)) = csc(nu*Pi)*(- sin(m*nu*Pi)*BesselK(nu, z*exp(- Pi*I))+ sin((m + 1)*nu*Pi)*BesselK(nu, z))
 
BesselK[\[Nu], z*Exp[m*Pi*I]] == Csc[\[Nu]*Pi]*(- Sin[m*\[Nu]*Pi]*BesselK[\[Nu], z*Exp[- Pi*I]]+ Sin[(m + 1)*\[Nu]*Pi]*BesselK[\[Nu], z])
 Failure Failure
Failed [170 / 210]
Result: 2.965939338+3.157233717*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 1}

Result: -10.37113929-12.75980866*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 2}

... skip entries to safe data
Failed [182 / 210]
Result: Complex[2.9659393403344363, 3.1572337219665294]
Test Values: {Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-10.371139260352981, -12.759808690998973]
Test Values: {Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.34.E5 K n ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = ( - 1 ) m ⁢ n ⁢ K n ⁡ ( z ) + ( - 1 ) n ⁢ ( m - 1 ) - 1 ⁢ m ⁢ π ⁢ i ⁢ I n ⁡ ( z ) modified-Bessel-second-kind 𝑛 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 superscript 1 𝑚 𝑛 modified-Bessel-second-kind 𝑛 𝑧 superscript 1 𝑛 𝑚 1 1 𝑚 𝜋 𝑖 modified-Bessel-first-kind 𝑛 𝑧 {\displaystyle{\displaystyle K_{n}\left(ze^{m\pi i}\right)=(-1)^{mn}K_{n}\left% (z\right)+(-1)^{n(m-1)-1}m\pi iI_{n}\left(z\right)}}
\modBesselK{n}@{ze^{m\pi i}} = (-1)^{mn}\modBesselK{n}@{z}+(-1)^{n(m-1)-1}m\pi i\modBesselI{n}@{z}		 ℜ ⁡ ( n + k + 1 ) > 0 𝑛 𝑘 1 0 {\displaystyle{\displaystyle\Re(n+k+1)>0}} 
BesselK(n, z*exp(m*Pi*I)) = (- 1)^(m*n)* BesselK(n, z)+(- 1)^(n*(m - 1)- 1)* m*Pi*I*BesselI(n, z)
 
BesselK[n, z*Exp[m*Pi*I]] == (- 1)^(m*n)* BesselK[n, z]+(- 1)^(n*(m - 1)- 1)* m*Pi*I*BesselI[n, z]
 Failure Failure
Failed [57 / 63]
Result: -1.971501919+2.706233555*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, m = 1, n = 1}

Result: -.7368261646+.3579119854*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, m = 1, n = 2}

... skip entries to safe data
Failed [48 / 63]
Result: Complex[-1.9715019183470535, 2.7062335550125516]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-0.736826162742255, 0.3579119863626685]
Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.34.E6 K n ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = + ( - 1 ) n ⁢ ( m - 1 ) ⁢ m ⁢ K n ⁡ ( z ⁢ e + π ⁢ i ) - ( - 1 ) n ⁢ m ⁢ ( m - 1 ) ⁢ K n ⁡ ( z ) modified-Bessel-second-kind 𝑛 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 superscript 1 𝑛 𝑚 1 𝑚 modified-Bessel-second-kind 𝑛 𝑧 superscript 𝑒 𝜋 𝑖 superscript 1 𝑛 𝑚 𝑚 1 modified-Bessel-second-kind 𝑛 𝑧 {\displaystyle{\displaystyle K_{n}\left(ze^{m\pi i}\right)=+(-1)^{n(m-1)}mK_{n% }\left(ze^{+\pi i}\right)-(-1)^{nm}(m-1)K_{n}\left(z\right)}}
\modBesselK{n}@{ze^{m\pi i}} = +(-1)^{n(m-1)}m\modBesselK{n}@{ze^{+\pi i}}-(-1)^{nm}(m- 1)\modBesselK{n}@{z}		 

BesselK(n, z*exp(m*Pi*I)) = +(- 1)^(n*(m - 1))* m*BesselK(n, z*exp(+ Pi*I))-(- 1)^(n*m)*(m - 1)*BesselK(n, z)
 
BesselK[n, z*Exp[m*Pi*I]] == +(- 1)^(n*(m - 1))* m*BesselK[n, z*Exp[+ Pi*I]]-(- 1)^(n*m)*(m - 1)*BesselK[n, z]
 Failure Failure
Failed [51 / 63]
Result: -1.971501920+2.706233556*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, m = 2, n = 1}

Result: .7368261602-.357911988*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, m = 2, n = 2}

... skip entries to safe data
Failed [42 / 63]
Result: Complex[-1.9715019183470535, 2.7062335550125516]
Test Values: {Rule[m, 2], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[0.736826162742255, -0.3579119863626685]
Test Values: {Rule[m, 2], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.34.E6 K n ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = - ( - 1 ) n ⁢ ( m - 1 ) ⁢ m ⁢ K n ⁡ ( z ⁢ e - π ⁢ i ) + ( - 1 ) n ⁢ m ⁢ ( m + 1 ) ⁢ K n ⁡ ( z ) modified-Bessel-second-kind 𝑛 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 superscript 1 𝑛 𝑚 1 𝑚 modified-Bessel-second-kind 𝑛 𝑧 superscript 𝑒 𝜋 𝑖 superscript 1 𝑛 𝑚 𝑚 1 modified-Bessel-second-kind 𝑛 𝑧 {\displaystyle{\displaystyle K_{n}\left(ze^{m\pi i}\right)=-(-1)^{n(m-1)}mK_{n% }\left(ze^{-\pi i}\right)+(-1)^{nm}(m+1)K_{n}\left(z\right)}}
\modBesselK{n}@{ze^{m\pi i}} = -(-1)^{n(m-1)}m\modBesselK{n}@{ze^{-\pi i}}+(-1)^{nm}(m+ 1)\modBesselK{n}@{z}		 

BesselK(n, z*exp(m*Pi*I)) = -(- 1)^(n*(m - 1))* m*BesselK(n, z*exp(- Pi*I))+(- 1)^(n*m)*(m + 1)*BesselK(n, z)
 
BesselK[n, z*Exp[m*Pi*I]] == -(- 1)^(n*(m - 1))* m*BesselK[n, z*Exp[- Pi*I]]+(- 1)^(n*m)*(m + 1)*BesselK[n, z]
 Failure Failure
Failed [54 / 63]
Result: -1.971501919+2.706233556*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, m = 1, n = 1}

Result: -.7368261645+.357911985*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, m = 1, n = 2}

... skip entries to safe data
Failed [63 / 63]
Result: Complex[-1.9715019183470535, 2.7062335550125516]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-0.736826162742255, 0.3579119863626685]
Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.34#Ex1 I ν ⁡ ( z ¯ ) = I ν ⁡ ( z ) ¯ modified-Bessel-first-kind 𝜈 𝑧 modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\nu}\left(\overline{z}\right)=\overline{I_{\nu% }\left(z\right)}}}
\modBesselI{\nu}@{\conj{z}} = \conj{\modBesselI{\nu}@{z}}		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
BesselI(nu, conjugate(z)) = conjugate(BesselI(nu, z))
 
BesselI[\[Nu], Conjugate[z]] == Conjugate[BesselI[\[Nu], z]]
 Failure Failure Skipped - Because timed out
Failed [28 / 70]
Result: Complex[-0.1457476573229447, -0.7449450592023206]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[1.100244133383339, 1.2347828003590728]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.34#Ex2 K ν ⁡ ( z ¯ ) = K ν ⁡ ( z ) ¯ modified-Bessel-second-kind 𝜈 𝑧 modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\nu}\left(\overline{z}\right)=\overline{K_{\nu% }\left(z\right)}}}
\modBesselK{\nu}@{\conj{z}} = \conj{\modBesselK{\nu}@{z}}		 

BesselK(nu, conjugate(z)) = conjugate(BesselK(nu, z))
 
BesselK[\[Nu], Conjugate[z]] == Conjugate[BesselK[\[Nu], z]]
 Failure Failure
Failed [28 / 70]
Result: -.3322466664+.1347267497*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}

Result: .8978926857-1.555608423*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [28 / 70]
Result: Complex[-0.332246666369582, 0.13472674975137633]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[0.23222824698313052, -0.12812607679285354]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.35.E1 e 1 2 ⁢ z ⁢ ( t + t - 1 ) = ∑ m = - ∞ ∞ t m ⁢ I m ⁡ ( z ) superscript 𝑒 1 2 𝑧 𝑡 superscript 𝑡 1 superscript subscript 𝑚 superscript 𝑡 𝑚 modified-Bessel-first-kind 𝑚 𝑧 {\displaystyle{\displaystyle e^{\frac{1}{2}z(t+t^{-1})}=\sum_{m=-\infty}^{% \infty}t^{m}I_{m}\left(z\right)}}
e^{\frac{1}{2}z(t+t^{-1})} = \sum_{m=-\infty}^{\infty}t^{m}\modBesselI{m}@{z}		 ℜ ⁡ ( m + k + 1 ) > 0 𝑚 𝑘 1 0 {\displaystyle{\displaystyle\Re(m+k+1)>0}} 
exp((1)/(2)*z*(t + (t)^(- 1))) = sum((t)^(m)* BesselI(m, z), m = - infinity..infinity)
 
Exp[Divide[1,2]*z*(t + (t)^(- 1))] == Sum[(t)^(m)* BesselI[m, z], {m, - Infinity, Infinity}, GenerateConditions->None]
 Failure Aborted Skipped - Because timed out Skipped - Because timed out
10.35.E2 e z ⁢ cos ⁡ θ = I 0 ⁡ ( z ) + 2 ⁢ ∑ k = 1 ∞ I k ⁡ ( z ) ⁢ cos ⁡ ( k ⁢ θ ) superscript 𝑒 𝑧 𝜃 modified-Bessel-first-kind 0 𝑧 2 superscript subscript 𝑘 1 modified-Bessel-first-kind 𝑘 𝑧 𝑘 𝜃 {\displaystyle{\displaystyle e^{z\cos\theta}=I_{0}\left(z\right)+2\sum_{k=1}^{% \infty}I_{k}\left(z\right)\cos\left(k\theta\right)}}
e^{z\cos@@{\theta}} = \modBesselI{0}@{z}+2\sum_{k=1}^{\infty}\modBesselI{k}@{z}\cos@{k\theta}		 ℜ ⁡ ( 0 + k + 1 ) > 0 , ℜ ⁡ ( k + k + 1 ) > 0 formulae-sequence 0 𝑘 1 0 𝑘 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(k+k+1)>0}} 
exp(z*cos(theta)) = BesselI(0, z)+ 2*sum(BesselI(k, z)*cos(k*theta), k = 1..infinity)
 
Exp[z*Cos[\[Theta]]] == BesselI[0, z]+ 2*Sum[BesselI[k, z]*Cos[k*\[Theta]], {k, 1, Infinity}, GenerateConditions->None]
 Failure Successful Skipped - Because timed out Successful [Tested: 70]
10.35.E3 e z ⁢ sin ⁡ θ = I 0 ⁡ ( z ) + 2 ⁢ ∑ k = 0 ∞ ( - 1 ) k ⁢ I 2 ⁢ k + 1 ⁡ ( z ) ⁢ sin ⁡ ( ( 2 ⁢ k + 1 ) ⁢ θ ) + 2 ⁢ ∑ k = 1 ∞ ( - 1 ) k ⁢ I 2 ⁢ k ⁡ ( z ) ⁢ cos ⁡ ( 2 ⁢ k ⁢ θ ) superscript 𝑒 𝑧 𝜃 modified-Bessel-first-kind 0 𝑧 2 superscript subscript 𝑘 0 superscript 1 𝑘 modified-Bessel-first-kind 2 𝑘 1 𝑧 2 𝑘 1 𝜃 2 superscript subscript 𝑘 1 superscript 1 𝑘 modified-Bessel-first-kind 2 𝑘 𝑧 2 𝑘 𝜃 {\displaystyle{\displaystyle e^{z\sin\theta}=I_{0}\left(z\right)+2\sum_{k=0}^{% \infty}(-1)^{k}I_{2k+1}\left(z\right)\sin\left((2k+1)\theta\right)+2\sum_{k=1}% ^{\infty}(-1)^{k}I_{2k}\left(z\right)\cos\left(2k\theta\right)}}
e^{z\sin@@{\theta}} = \modBesselI{0}@{z}+2\sum_{k=0}^{\infty}(-1)^{k}\modBesselI{2k+1}@{z}\sin@{(2k+1)\theta}+2\sum_{k=1}^{\infty}(-1)^{k}\modBesselI{2k}@{z}\cos@{2k\theta}		 ℜ ⁡ ( 0 + k + 1 ) > 0 , ℜ ⁡ ( ( 2 ⁢ k + 1 ) + k + 1 ) > 0 , ℜ ⁡ ( ( 2 ⁢ k ) + k + 1 ) > 0 formulae-sequence 0 𝑘 1 0 formulae-sequence 2 𝑘 1 𝑘 1 0 2 𝑘 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0,\Re((2k+1)+k+1)>0,\Re((2k)+k+1)>0}} 
exp(z*sin(theta)) = BesselI(0, z)+ 2*sum((- 1)^(k)* BesselI(2*k + 1, z)*sin((2*k + 1)*theta), k = 0..infinity)+ 2*sum((- 1)^(k)* BesselI(2*k, z)*cos(2*k*theta), k = 1..infinity)
 
Exp[z*Sin[\[Theta]]] == BesselI[0, z]+ 2*Sum[(- 1)^(k)* BesselI[2*k + 1, z]*Sin[(2*k + 1)*\[Theta]], {k, 0, Infinity}, GenerateConditions->None]+ 2*Sum[(- 1)^(k)* BesselI[2*k, z]*Cos[2*k*\[Theta]], {k, 1, Infinity}, GenerateConditions->None]
 Aborted Failure Manual Skip! Skipped - Because timed out
10.35.E4 1 = I 0 ⁡ ( z ) - 2 ⁢ I 2 ⁡ ( z ) + 2 ⁢ I 4 ⁡ ( z ) - 2 ⁢ I 6 ⁡ ( z ) + ⋯ 1 modified-Bessel-first-kind 0 𝑧 2 modified-Bessel-first-kind 2 𝑧 2 modified-Bessel-first-kind 4 𝑧 2 modified-Bessel-first-kind 6 𝑧 ⋯ {\displaystyle{\displaystyle 1=I_{0}\left(z\right)-2I_{2}\left(z\right)+2I_{4}% \left(z\right)-2I_{6}\left(z\right)+\cdots}}
1 = \modBesselI{0}@{z}-2\modBesselI{2}@{z}+2\modBesselI{4}@{z}-2\modBesselI{6}@{z}+\dotsb		 ℜ ⁡ ( 0 + k + 1 ) > 0 , ℜ ⁡ ( 2 + k + 1 ) > 0 , ℜ ⁡ ( 4 + k + 1 ) > 0 , ℜ ⁡ ( 6 + k + 1 ) > 0 formulae-sequence 0 𝑘 1 0 formulae-sequence 2 𝑘 1 0 formulae-sequence 4 𝑘 1 0 6 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(2+k+1)>0,\Re(4+k+1)>0,\Re(6+k+1)>% 0}} 
1 = BesselI(0, z)- 2*BesselI(2, z)+ 2*BesselI(4, z)- 2*BesselI(6, z)+ ..
 
1 == BesselI[0, z]- 2*BesselI[2, z]+ 2*BesselI[4, z]- 2*BesselI[6, z]+ \[Ellipsis]
 Error Failure -
Failed [7 / 7]
Result: Plus[Complex[-9.440290591519046*^-8, -1.7199789187696823*^-7], Times[-1.0, …]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Plus[Complex[-9.924736610669727*^-8, -1.6360842739013975*^-7], Times[-1.0, …]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.35.E5 e + z = I 0 ⁡ ( z ) + 2 ⁢ I 1 ⁡ ( z ) + 2 ⁢ I 2 ⁡ ( z ) + 2 ⁢ I 3 ⁡ ( z ) + ⋯ superscript 𝑒 𝑧 modified-Bessel-first-kind 0 𝑧 2 modified-Bessel-first-kind 1 𝑧 2 modified-Bessel-first-kind 2 𝑧 2 modified-Bessel-first-kind 3 𝑧 ⋯ {\displaystyle{\displaystyle e^{+z}=I_{0}\left(z\right)+2I_{1}\left(z\right)+2% I_{2}\left(z\right)+2I_{3}\left(z\right)+\cdots}}
e^{+ z} = \modBesselI{0}@{z}+ 2\modBesselI{1}@{z}+2\modBesselI{2}@{z}+ 2\modBesselI{3}@{z}+\dotsb		 ℜ ⁡ ( 0 + k + 1 ) > 0 , ℜ ⁡ ( 1 + k + 1 ) > 0 , ℜ ⁡ ( 2 + k + 1 ) > 0 , ℜ ⁡ ( 3 + k + 1 ) > 0 formulae-sequence 0 𝑘 1 0 formulae-sequence 1 𝑘 1 0 formulae-sequence 2 𝑘 1 0 3 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(1+k+1)>0,\Re(2+k+1)>0,\Re(3+k+1)>% 0}} 
exp(+ z) = BesselI(0, z)+ 2*BesselI(1, z)+ 2*BesselI(2, z)+ 2*BesselI(3, z)+ ..
 
Exp[+ z] == BesselI[0, z]+ 2*BesselI[1, z]+ 2*BesselI[2, z]+ 2*BesselI[3, z]+ \[Ellipsis]
 Error Failure -
Failed [7 / 7]
Result: Plus[Complex[-0.003384051289485407, 0.00475177611436145], Times[-1.0, …]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Plus[Complex[-0.002576303532707505, 0.004074841322498801], Times[-1.0, …]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.35.E5 e - z = I 0 ⁡ ( z ) - 2 ⁢ I 1 ⁡ ( z ) + 2 ⁢ I 2 ⁡ ( z ) - 2 ⁢ I 3 ⁡ ( z ) + ⋯ superscript 𝑒 𝑧 modified-Bessel-first-kind 0 𝑧 2 modified-Bessel-first-kind 1 𝑧 2 modified-Bessel-first-kind 2 𝑧 2 modified-Bessel-first-kind 3 𝑧 ⋯ {\displaystyle{\displaystyle e^{-z}=I_{0}\left(z\right)-2I_{1}\left(z\right)+2% I_{2}\left(z\right)-2I_{3}\left(z\right)+\cdots}}
e^{- z} = \modBesselI{0}@{z}- 2\modBesselI{1}@{z}+2\modBesselI{2}@{z}- 2\modBesselI{3}@{z}+\dotsb		 ℜ ⁡ ( 0 + k + 1 ) > 0 , ℜ ⁡ ( 1 + k + 1 ) > 0 , ℜ ⁡ ( 2 + k + 1 ) > 0 , ℜ ⁡ ( 3 + k + 1 ) > 0 formulae-sequence 0 𝑘 1 0 formulae-sequence 1 𝑘 1 0 formulae-sequence 2 𝑘 1 0 3 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(1+k+1)>0,\Re(2+k+1)>0,\Re(3+k+1)>% 0}} 
exp(- z) = BesselI(0, z)- 2*BesselI(1, z)+ 2*BesselI(2, z)- 2*BesselI(3, z)+ ..
 
Exp[- z] == BesselI[0, z]- 2*BesselI[1, z]+ 2*BesselI[2, z]- 2*BesselI[3, z]+ \[Ellipsis]
 Error Failure -
Failed [7 / 7]
Result: Plus[Complex[-0.0024389937896763803, 0.0042567403420422645], Times[-1.0, …]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Plus[Complex[-0.0020316532349716754, 0.004934003265463338], Times[-1.0, …]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.37.E1 | K ν ⁡ ( z ) | < | K μ ⁡ ( z ) | modified-Bessel-second-kind 𝜈 𝑧 modified-Bessel-second-kind 𝜇 𝑧 {\displaystyle{\displaystyle|K_{\nu}\left(z\right)|<|K_{\mu}\left(z\right)|}}
|\modBesselK{\nu}@{z}| < |\modBesselK{\mu}@{z}|		 

abs(BesselK(nu, z)) < abs(BesselK(mu, z))
 
Abs[BesselK[\[Nu], z]] < Abs[BesselK[\[Mu], z]]
 Failure Failure
Failed [204 / 300]
Result: .6496143723 < .6496143723
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}

Result: 3.110500858 < 3.110500858
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [184 / 300]
Result: False
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: False
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
10.38.E1 ∂ ⁡ I + ν ⁡ ( z ) ∂ ⁡ ν = + I + ν ⁡ ( z ) ⁢ ln ⁡ ( 1 2 ⁢ z ) - ( 1 2 ⁢ z ) + ν ⁢ ∑ k = 0 ∞ ψ ⁡ ( k + 1 + ν ) Γ ⁡ ( k + 1 + ν ) ⁢ ( 1 4 ⁢ z 2 ) k k ! partial-derivative modified-Bessel-first-kind 𝜈 𝑧 𝜈 modified-Bessel-first-kind 𝜈 𝑧 1 2 𝑧 superscript 1 2 𝑧 𝜈 superscript subscript 𝑘 0 digamma 𝑘 1 𝜈 Euler-Gamma 𝑘 1 𝜈 superscript 1 4 superscript 𝑧 2 𝑘 𝑘 {\displaystyle{\displaystyle\frac{\partial I_{+\nu}\left(z\right)}{\partial\nu% }=+I_{+\nu}\left(z\right)\ln\left(\tfrac{1}{2}z\right)-(\tfrac{1}{2}z)^{+\nu}% \sum_{k=0}^{\infty}\frac{\psi\left(k+1+\nu\right)}{\Gamma\left(k+1+\nu\right)}% \frac{(\frac{1}{4}z^{2})^{k}}{k!}}}
\pderiv{\modBesselI{+\nu}@{z}}{\nu} = +\modBesselI{+\nu}@{z}\ln@{\tfrac{1}{2}z}-(\tfrac{1}{2}z)^{+\nu}\sum_{k=0}^{\infty}\frac{\digamma@{k+1+\nu}}{\EulerGamma@{k+1+\nu}}\frac{(\frac{1}{4}z^{2})^{k}}{k!}		 ℜ ⁡ ( k + 1 + ν ) > 0 𝑘 1 𝜈 0 {\displaystyle{\displaystyle\Re(k+1+\nu)>0}} 
diff(BesselI(+ nu, z), nu) = + BesselI(+ nu, z)*ln((1)/(2)*z)-((1)/(2)*z)^(+ nu)* sum((Psi(k + 1 + nu))/(GAMMA(k + 1 + nu))*(((1)/(4)*(z)^(2))^(k))/(factorial(k)), k = 0..infinity)
 
D[BesselI[+ \[Nu], z], \[Nu]] == + BesselI[+ \[Nu], z]*Log[Divide[1,2]*z]-(Divide[1,2]*z)^(+ \[Nu])* Sum[Divide[PolyGamma[k + 1 + \[Nu]],Gamma[k + 1 + \[Nu]]]*Divide[(Divide[1,4]*(z)^(2))^(k),(k)!], {k, 0, Infinity}, GenerateConditions->None]
 Failure Failure Skipped - Because timed out
Failed [7 / 70]
Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]}

Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, -2]}

... skip entries to safe data
10.38.E1 ∂ ⁡ I - ν ⁡ ( z ) ∂ ⁡ ν = - I - ν ⁡ ( z ) ⁢ ln ⁡ ( 1 2 ⁢ z ) + ( 1 2 ⁢ z ) - ν ⁢ ∑ k = 0 ∞ ψ ⁡ ( k + 1 - ν ) Γ ⁡ ( k + 1 - ν ) ⁢ ( 1 4 ⁢ z 2 ) k k ! partial-derivative modified-Bessel-first-kind 𝜈 𝑧 𝜈 modified-Bessel-first-kind 𝜈 𝑧 1 2 𝑧 superscript 1 2 𝑧 𝜈 superscript subscript 𝑘 0 digamma 𝑘 1 𝜈 Euler-Gamma 𝑘 1 𝜈 superscript 1 4 superscript 𝑧 2 𝑘 𝑘 {\displaystyle{\displaystyle\frac{\partial I_{-\nu}\left(z\right)}{\partial\nu% }=-I_{-\nu}\left(z\right)\ln\left(\tfrac{1}{2}z\right)+(\tfrac{1}{2}z)^{-\nu}% \sum_{k=0}^{\infty}\frac{\psi\left(k+1-\nu\right)}{\Gamma\left(k+1-\nu\right)}% \frac{(\frac{1}{4}z^{2})^{k}}{k!}}}
\pderiv{\modBesselI{-\nu}@{z}}{\nu} = -\modBesselI{-\nu}@{z}\ln@{\tfrac{1}{2}z}+(\tfrac{1}{2}z)^{-\nu}\sum_{k=0}^{\infty}\frac{\digamma@{k+1-\nu}}{\EulerGamma@{k+1-\nu}}\frac{(\frac{1}{4}z^{2})^{k}}{k!}		 ℜ ⁡ ( k + 1 + ν ) > 0 , ℜ ⁡ ( k + 1 - ν ) > 0 , ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 formulae-sequence 𝑘 1 𝜈 0 formulae-sequence 𝑘 1 𝜈 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(k+1+\nu)>0,\Re(k+1-\nu)>0,\Re((-\nu)+k+1)>0}} 
diff(BesselI(- nu, z), nu) = - BesselI(- nu, z)*ln((1)/(2)*z)+((1)/(2)*z)^(- nu)* sum((Psi(k + 1 - nu))/(GAMMA(k + 1 - nu))*(((1)/(4)*(z)^(2))^(k))/(factorial(k)), k = 0..infinity)
 
D[BesselI[- \[Nu], z], \[Nu]] == - BesselI[- \[Nu], z]*Log[Divide[1,2]*z]+(Divide[1,2]*z)^(- \[Nu])* Sum[Divide[PolyGamma[k + 1 - \[Nu]],Gamma[k + 1 - \[Nu]]]*Divide[(Divide[1,4]*(z)^(2))^(k),(k)!], {k, 0, Infinity}, GenerateConditions->None]
 Failure Failure Skipped - Because timed out
Failed [7 / 70]
Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]}

Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, 2]}

... skip entries to safe data
10.38.E2 ∂ ⁡ K ν ⁡ ( z ) ∂ ⁡ ν = 1 2 ⁢ π ⁢ csc ⁡ ( ν ⁢ π ) ⁢ ( ∂ ⁡ I - ν ⁡ ( z ) ∂ ⁡ ν - ∂ ⁡ I ν ⁡ ( z ) ∂ ⁡ ν ) - π ⁢ cot ⁡ ( ν ⁢ π ) ⁢ K ν ⁡ ( z ) partial-derivative modified-Bessel-second-kind 𝜈 𝑧 𝜈 1 2 𝜋 𝜈 𝜋 partial-derivative modified-Bessel-first-kind 𝜈 𝑧 𝜈 partial-derivative modified-Bessel-first-kind 𝜈 𝑧 𝜈 𝜋 𝜈 𝜋 modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle\frac{\partial K_{\nu}\left(z\right)}{\partial\nu}% =\tfrac{1}{2}\pi\csc\left(\nu\pi\right)\*\left(\frac{\partial I_{-\nu}\left(z% \right)}{\partial\nu}-\frac{\partial I_{\nu}\left(z\right)}{\partial\nu}\right% )-\pi\cot\left(\nu\pi\right)K_{\nu}\left(z\right)}}
\pderiv{\modBesselK{\nu}@{z}}{\nu} = \tfrac{1}{2}\pi\csc@{\nu\pi}\*\left(\pderiv{\modBesselI{-\nu}@{z}}{\nu}-\pderiv{\modBesselI{\nu}@{z}}{\nu}\right)-\pi\cot@{\nu\pi}\modBesselK{\nu}@{z}		 ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re((-\nu)+k+1)>0,\Re(\nu+k+1)>0}} 
diff(BesselK(nu, z), nu) = (1)/(2)*Pi*csc(nu*Pi)*(diff(BesselI(- nu, z), nu)- diff(BesselI(nu, z), nu))- Pi*cot(nu*Pi)*BesselK(nu, z)
 
D[BesselK[\[Nu], z], \[Nu]] == Divide[1,2]*Pi*Csc[\[Nu]*Pi]*(D[BesselI[- \[Nu], z], \[Nu]]- D[BesselI[\[Nu], z], \[Nu]])- Pi*Cot[\[Nu]*Pi]*BesselK[\[Nu], z]
 Successful Failure - Successful [Tested: 7]
10.39#Ex1 I 1 2 ⁡ ( z ) = ( 2 π ⁢ z ) 1 2 ⁢ sinh ⁡ z modified-Bessel-first-kind 1 2 𝑧 superscript 2 𝜋 𝑧 1 2 𝑧 {\displaystyle{\displaystyle I_{\frac{1}{2}}\left(z\right)=\left(\frac{2}{\pi z% }\right)^{\frac{1}{2}}\sinh z}}
\modBesselI{\frac{1}{2}}@{z} = \left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\sinh@@{z}		 ℜ ⁡ ( ( 1 2 ) + k + 1 ) > 0 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((\frac{1}{2})+k+1)>0}} 
BesselI((1)/(2), z) = ((2)/(Pi*z))^((1)/(2))* sinh(z)
 
BesselI[Divide[1,2], z] == (Divide[2,Pi*z])^(Divide[1,2])* Sinh[z]
 Failure Failure Successful [Tested: 7] Successful [Tested: 7]
10.39#Ex2 I - 1 2 ⁡ ( z ) = ( 2 π ⁢ z ) 1 2 ⁢ cosh ⁡ z modified-Bessel-first-kind 1 2 𝑧 superscript 2 𝜋 𝑧 1 2 𝑧 {\displaystyle{\displaystyle I_{-\frac{1}{2}}\left(z\right)=\left(\frac{2}{\pi z% }\right)^{\frac{1}{2}}\cosh z}}
\modBesselI{-\frac{1}{2}}@{z} = \left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\cosh@@{z}		 ℜ ⁡ ( ( - 1 2 ) + k + 1 ) > 0 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((-\frac{1}{2})+k+1)>0}} 
BesselI(-(1)/(2), z) = ((2)/(Pi*z))^((1)/(2))* cosh(z)
 
BesselI[-Divide[1,2], z] == (Divide[2,Pi*z])^(Divide[1,2])* Cosh[z]
 Failure Failure Successful [Tested: 7] Successful [Tested: 7]
10.39.E2 K 1 2 ⁡ ( z ) = K - 1 2 ⁡ ( z ) modified-Bessel-second-kind 1 2 𝑧 modified-Bessel-second-kind 1 2 𝑧 {\displaystyle{\displaystyle K_{\frac{1}{2}}\left(z\right)=K_{-\frac{1}{2}}% \left(z\right)}}
\modBesselK{\frac{1}{2}}@{z} = \modBesselK{-\frac{1}{2}}@{z}		 

BesselK((1)/(2), z) = BesselK(-(1)/(2), z)
 
BesselK[Divide[1,2], z] == BesselK[-Divide[1,2], z]
 Successful Successful Skip - symbolical successful subtest Successful [Tested: 7]
10.39.E2 K - 1 2 ⁡ ( z ) = ( π 2 ⁢ z ) 1 2 ⁢ e - z modified-Bessel-second-kind 1 2 𝑧 superscript 𝜋 2 𝑧 1 2 superscript 𝑒 𝑧 {\displaystyle{\displaystyle K_{-\frac{1}{2}}\left(z\right)=\left(\frac{\pi}{2% z}\right)^{\frac{1}{2}}e^{-z}}}
\modBesselK{-\frac{1}{2}}@{z} = \left(\frac{\pi}{2z}\right)^{\frac{1}{2}}e^{-z}		 

BesselK(-(1)/(2), z) = ((Pi)/(2*z))^((1)/(2))* exp(- z)
 
BesselK[-Divide[1,2], z] == (Divide[Pi,2*z])^(Divide[1,2])* Exp[- z]
 Failure Failure Successful [Tested: 7] Successful [Tested: 7]
10.39.E3 K 1 4 ⁡ ( z ) = π 1 2 ⁢ z - 1 4 ⁢ U ⁡ ( 0 , 2 ⁢ z 1 2 ) modified-Bessel-second-kind 1 4 𝑧 superscript 𝜋 1 2 superscript 𝑧 1 4 parabolic-U 0 2 superscript 𝑧 1 2 {\displaystyle{\displaystyle K_{\frac{1}{4}}\left(z\right)=\pi^{\frac{1}{2}}z^% {-\frac{1}{4}}U\left(0,2z^{\frac{1}{2}}\right)}}
\modBesselK{\frac{1}{4}}@{z} = \pi^{\frac{1}{2}}z^{-\frac{1}{4}}\paraU@{0}{2z^{\frac{1}{2}}}		 

BesselK((1)/(4), z) = (Pi)^((1)/(2))* (z)^(-(1)/(4))* CylinderU(0, 2*(z)^((1)/(2)))
 
BesselK[Divide[1,4], z] == (Pi)^(Divide[1,2])* (z)^(-Divide[1,4])* ParabolicCylinderD[- 1/2 -(0), 2*(z)^(Divide[1,2])]
 Successful Failure - Successful [Tested: 7]
10.39.E4 K 3 4 ⁡ ( z ) = 1 2 ⁢ π 1 2 ⁢ z - 3 4 ⁢ ( 1 2 ⁢ U ⁡ ( 1 , 2 ⁢ z 1 2 ) + U ⁡ ( - 1 , 2 ⁢ z 1 2 ) ) modified-Bessel-second-kind 3 4 𝑧 1 2 superscript 𝜋 1 2 superscript 𝑧 3 4 1 2 parabolic-U 1 2 superscript 𝑧 1 2 parabolic-U 1 2 superscript 𝑧 1 2 {\displaystyle{\displaystyle K_{\frac{3}{4}}\left(z\right)=\tfrac{1}{2}\pi^{% \frac{1}{2}}z^{-\frac{3}{4}}\left(\tfrac{1}{2}U\left(1,2z^{\frac{1}{2}}\right)% +U\left(-1,2z^{\frac{1}{2}}\right)\right)}}
\modBesselK{\frac{3}{4}}@{z} = \tfrac{1}{2}\pi^{\frac{1}{2}}z^{-\frac{3}{4}}\left(\tfrac{1}{2}\paraU@{1}{2z^{\frac{1}{2}}}+\paraU@{-1}{2z^{\frac{1}{2}}}\right)		 

BesselK((3)/(4), z) = (1)/(2)*(Pi)^((1)/(2))* (z)^(-(3)/(4))*((1)/(2)*CylinderU(1, 2*(z)^((1)/(2)))+ CylinderU(- 1, 2*(z)^((1)/(2))))
 
BesselK[Divide[3,4], z] == Divide[1,2]*(Pi)^(Divide[1,2])* (z)^(-Divide[3,4])*(Divide[1,2]*ParabolicCylinderD[- 1/2 -(1), 2*(z)^(Divide[1,2])]+ ParabolicCylinderD[- 1/2 -(- 1), 2*(z)^(Divide[1,2])])
 Failure Failure Successful [Tested: 7] Successful [Tested: 7]
10.39.E5 I ν ⁡ ( z ) = ( 1 2 ⁢ z ) ν ⁢ e + z Γ ⁡ ( ν + 1 ) ⁢ M ⁡ ( ν + 1 2 , 2 ⁢ ν + 1 , - 2 ⁢ z ) modified-Bessel-first-kind 𝜈 𝑧 superscript 1 2 𝑧 𝜈 superscript 𝑒 𝑧 Euler-Gamma 𝜈 1 Kummer-confluent-hypergeometric-M 𝜈 1 2 2 𝜈 1 2 𝑧 {\displaystyle{\displaystyle I_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}% e^{+z}}{\Gamma\left(\nu+1\right)}M\left(\nu+\tfrac{1}{2},2\nu+1,-2z\right)}}
\modBesselI{\nu}@{z} = \frac{(\tfrac{1}{2}z)^{\nu}e^{+ z}}{\EulerGamma@{\nu+1}}\KummerconfhyperM@{\nu+\tfrac{1}{2}}{2\nu+1}{- 2z}		 ℜ ⁡ ( ν + 1 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+1)>0,\Re(\nu+k+1)>0}} 
BesselI(nu, z) = (((1)/(2)*z)^(nu)* exp(+ z))/(GAMMA(nu + 1))*KummerM(nu +(1)/(2), 2*nu + 1, - 2*z)
 
BesselI[\[Nu], z] == Divide[(Divide[1,2]*z)^\[Nu]* Exp[+ z],Gamma[\[Nu]+ 1]]*Hypergeometric1F1[\[Nu]+Divide[1,2], 2*\[Nu]+ 1, - 2*z]
 Failure Successful
Failed [7 / 56]
Result: -.800260207-.3396157390*I
Test Values: {nu = -1/2, z = 1/2*3^(1/2)+1/2*I}

Result: -.4588638571-.5759587792*I
Test Values: {nu = -1/2, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [7 / 56]
Result: Complex[-0.8002602062152042, -0.3396157389151986]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -0.5]}

Result: Complex[-0.45886385712966904, -0.5759587792371148]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, -0.5]}

... skip entries to safe data
10.39.E5 I ν ⁡ ( z ) = ( 1 2 ⁢ z ) ν ⁢ e - z Γ ⁡ ( ν + 1 ) ⁢ M ⁡ ( ν + 1 2 , 2 ⁢ ν + 1 , + 2 ⁢ z ) modified-Bessel-first-kind 𝜈 𝑧 superscript 1 2 𝑧 𝜈 superscript 𝑒 𝑧 Euler-Gamma 𝜈 1 Kummer-confluent-hypergeometric-M 𝜈 1 2 2 𝜈 1 2 𝑧 {\displaystyle{\displaystyle I_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}% e^{-z}}{\Gamma\left(\nu+1\right)}M\left(\nu+\tfrac{1}{2},2\nu+1,+2z\right)}}
\modBesselI{\nu}@{z} = \frac{(\tfrac{1}{2}z)^{\nu}e^{- z}}{\EulerGamma@{\nu+1}}\KummerconfhyperM@{\nu+\tfrac{1}{2}}{2\nu+1}{+ 2z}		 ℜ ⁡ ( ν + 1 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+1)>0,\Re(\nu+k+1)>0}} 
BesselI(nu, z) = (((1)/(2)*z)^(nu)* exp(- z))/(GAMMA(nu + 1))*KummerM(nu +(1)/(2), 2*nu + 1, + 2*z)
 
BesselI[\[Nu], z] == Divide[(Divide[1,2]*z)^\[Nu]* Exp[- z],Gamma[\[Nu]+ 1]]*Hypergeometric1F1[\[Nu]+Divide[1,2], 2*\[Nu]+ 1, + 2*z]
 Successful Successful Skip - symbolical successful subtest
Failed [7 / 56]
Result: Complex[0.8002602062152032, 0.3396157389151989]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -0.5]}

Result: Complex[0.4588638571296689, 0.575958779237115]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, -0.5]}

... skip entries to safe data
10.39.E6 K ν ⁡ ( z ) = π 1 2 ⁢ ( 2 ⁢ z ) ν ⁢ e - z ⁢ U ⁡ ( ν + 1 2 , 2 ⁢ ν + 1 , 2 ⁢ z ) modified-Bessel-second-kind 𝜈 𝑧 superscript 𝜋 1 2 superscript 2 𝑧 𝜈 superscript 𝑒 𝑧 Kummer-confluent-hypergeometric-U 𝜈 1 2 2 𝜈 1 2 𝑧 {\displaystyle{\displaystyle K_{\nu}\left(z\right)=\pi^{\frac{1}{2}}(2z)^{\nu}% e^{-z}U\left(\nu+\tfrac{1}{2},2\nu+1,2z\right)}}
\modBesselK{\nu}@{z} = \pi^{\frac{1}{2}}(2z)^{\nu}e^{-z}\KummerconfhyperU@{\nu+\tfrac{1}{2}}{2\nu+1}{2z}		 

BesselK(nu, z) = (Pi)^((1)/(2))*(2*z)^(nu)* exp(- z)*KummerU(nu +(1)/(2), 2*nu + 1, 2*z)
 
BesselK[\[Nu], z] == (Pi)^(Divide[1,2])*(2*z)^\[Nu]* Exp[- z]*HypergeometricU[\[Nu]+Divide[1,2], 2*\[Nu]+ 1, 2*z]
 Successful Successful - Successful [Tested: 70]
10.39.E7 I ν ⁡ ( z ) = ( 2 ⁢ z ) - 1 2 ⁢ M 0 , ν ⁡ ( 2 ⁢ z ) 2 2 ⁢ ν ⁢ Γ ⁡ ( ν + 1 ) modified-Bessel-first-kind 𝜈 𝑧 superscript 2 𝑧 1 2 Whittaker-confluent-hypergeometric-M 0 𝜈 2 𝑧 superscript 2 2 𝜈 Euler-Gamma 𝜈 1 {\displaystyle{\displaystyle I_{\nu}\left(z\right)=\frac{(2z)^{-\frac{1}{2}}M_% {0,\nu}\left(2z\right)}{2^{2\nu}\Gamma\left(\nu+1\right)}}}
\modBesselI{\nu}@{z} = \frac{(2z)^{-\frac{1}{2}}\WhittakerconfhyperM{0}{\nu}@{2z}}{2^{2\nu}\EulerGamma@{\nu+1}}		 ℜ ⁡ ( ν + 1 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+1)>0,\Re(\nu+k+1)>0}} 
BesselI(nu, z) = ((2*z)^(-(1)/(2))* WhittakerM(0, nu, 2*z))/((2)^(2*nu)* GAMMA(nu + 1))
 
BesselI[\[Nu], z] == Divide[(2*z)^(-Divide[1,2])* WhittakerM[0, \[Nu], 2*z],(2)^(2*\[Nu])* Gamma[\[Nu]+ 1]]
 Successful Successful - Successful [Tested: 7]
10.39.E8 K ν ⁡ ( z ) = ( π 2 ⁢ z ) 1 2 ⁢ W 0 , ν ⁡ ( 2 ⁢ z ) modified-Bessel-second-kind 𝜈 𝑧 superscript 𝜋 2 𝑧 1 2 Whittaker-confluent-hypergeometric-W 0 𝜈 2 𝑧 {\displaystyle{\displaystyle K_{\nu}\left(z\right)=\left(\frac{\pi}{2z}\right)% ^{\frac{1}{2}}W_{0,\nu}\left(2z\right)}}
\modBesselK{\nu}@{z} = \left(\frac{\pi}{2z}\right)^{\frac{1}{2}}\WhittakerconfhyperW{0}{\nu}@{2z}		 

BesselK(nu, z) = ((Pi)/(2*z))^((1)/(2))* WhittakerW(0, nu, 2*z)
 
BesselK[\[Nu], z] == (Divide[Pi,2*z])^(Divide[1,2])* WhittakerW[0, \[Nu], 2*z]
 Failure Failure Successful [Tested: 70] Successful [Tested: 70]
10.39.E9 I ν ⁡ ( z ) = ( 1 2 ⁢ z ) ν Γ ⁡ ( ν + 1 ) ⁢ F 1 0 ⁡ ( - ; ν + 1 ; 1 4 ⁢ z 2 ) modified-Bessel-first-kind 𝜈 𝑧 superscript 1 2 𝑧 𝜈 Euler-Gamma 𝜈 1 Gauss-hypergeometric-pFq 0 1 𝜈 1 1 4 superscript 𝑧 2 {\displaystyle{\displaystyle I_{\nu}\left(z\right)=\frac{(\frac{1}{2}z)^{\nu}}% {\Gamma\left(\nu+1\right)}{{}_{0}F_{1}}\left(-;\nu+1;\tfrac{1}{4}z^{2}\right)}}
\modBesselI{\nu}@{z} = \frac{(\frac{1}{2}z)^{\nu}}{\EulerGamma@{\nu+1}}\genhyperF{0}{1}@{-}{\nu+1}{\tfrac{1}{4}z^{2}}		 ℜ ⁡ ( ν + 1 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+1)>0,\Re(\nu+k+1)>0}} 
BesselI(nu, z) = (((1)/(2)*z)^(nu))/(GAMMA(nu + 1))*hypergeom([-], [nu + 1], (1)/(4)*(z)^(2))
 
BesselI[\[Nu], z] == Divide[(Divide[1,2]*z)^\[Nu],Gamma[\[Nu]+ 1]]*HypergeometricPFQ[{-}, {\[Nu]+ 1}, Divide[1,4]*(z)^(2)]
 Error Failure - Error
10.40.E10 K ν ⁡ ( z ) = ( π 2 ⁢ z ) 1 2 ⁢ e - z ⁢ ( ∑ k = 0 ℓ - 1 a k ⁢ ( ν ) z k + R ℓ ⁢ ( ν , z ) ) modified-Bessel-second-kind 𝜈 𝑧 superscript 𝜋 2 𝑧 1 2 superscript 𝑒 𝑧 superscript subscript 𝑘 0 ℓ 1 subscript 𝑎 𝑘 𝜈 superscript 𝑧 𝑘 subscript 𝑅 ℓ 𝜈 𝑧 {\displaystyle{\displaystyle K_{\nu}\left(z\right)=\left(\frac{\pi}{2z}\right)% ^{\frac{1}{2}}e^{-z}\left(\sum_{k=0}^{\ell-1}\frac{a_{k}(\nu)}{z^{k}}+R_{\ell}% (\nu,z)\right)}}
\modBesselK{\nu}@{z} = \left(\frac{\pi}{2z}\right)^{\frac{1}{2}}e^{-z}\left(\sum_{k=0}^{\ell-1}\frac{a_{k}(\nu)}{z^{k}}+R_{\ell}(\nu,z)\right)		 k ≥ 1 𝑘 1 {\displaystyle{\displaystyle k\geq 1}} 
BesselK(nu, z) = ((Pi)/(2*z))^((1)/(2))* exp(- z)*(sum((((4*(nu)^(2)- (1)^(2))*(4*(nu)^(2)- (3)^(2)) .. (4*(nu)^(2)-(2*k - 1)^(2)))/(factorial(k)*(8)^(k)))/((z)^(k)), k = 0..ell - 1)+ R[ell](nu , z))
 
BesselK[\[Nu], z] == (Divide[Pi,2*z])^(Divide[1,2])* Exp[- z]*(Sum[Divide[Divide[(4*\[Nu]^(2)- (1)^(2))*(4*\[Nu]^(2)- (3)^(2)) \[Ellipsis](4*\[Nu]^(2)-(2*k - 1)^(2)),(k)!*(8)^(k)],(z)^(k)], {k, 0, \[ScriptL]- 1}, GenerateConditions->None]+ Subscript[R, \[ScriptL]][\[Nu], z])
 Failure Failure Error Error
10.41.E8 p = ( 1 + z 2 ) - 1 2 𝑝 superscript 1 superscript 𝑧 2 1 2 {\displaystyle{\displaystyle p=(1+z^{2})^{-\frac{1}{2}}}}
p = (1+z^{2})^{-\frac{1}{2}}		 

p = (1 + (z)^(2))^(-(1)/(2))
 
p == (1 + (z)^(2))^(-Divide[1,2])
 Skipped - no semantic math Skipped - no semantic math - -
10.41#Ex3 U 1 ⁢ ( p ) = 1 24 ⁢ ( 3 ⁢ p - 5 ⁢ p 3 ) subscript 𝑈 1 𝑝 1 24 3 𝑝 5 superscript 𝑝 3 {\displaystyle{\displaystyle U_{1}(p)=\tfrac{1}{24}(3p-5p^{3})}}
U_{1}(p) = \tfrac{1}{24}(3p-5p^{3})		 

U[1](p) = (1)/(24)*(3*p - 5*(p)^(3))
 
Subscript[U, 1][p] == Divide[1,24]*(3*p - 5*(p)^(3))
 Skipped - no semantic math Skipped - no semantic math - -
10.41#Ex4 U 2 ⁢ ( p ) = 1 1152 ⁢ ( 81 ⁢ p 2 - 462 ⁢ p 4 + 385 ⁢ p 6 ) subscript 𝑈 2 𝑝 1 1152 81 superscript 𝑝 2 462 superscript 𝑝 4 385 superscript 𝑝 6 {\displaystyle{\displaystyle U_{2}(p)=\tfrac{1}{1152}(81p^{2}-462p^{4}+385p^{6% })}}
U_{2}(p) = \tfrac{1}{1152}(81p^{2}-462p^{4}+385p^{6})		 

U[2](p) = (1)/(1152)*(81*(p)^(2)- 462*(p)^(4)+ 385*(p)^(6))
 
Subscript[U, 2][p] == Divide[1,1152]*(81*(p)^(2)- 462*(p)^(4)+ 385*(p)^(6))
 Skipped - no semantic math Skipped - no semantic math - -
10.41#Ex5 U 3 ⁢ ( p ) = 1 4 14720 ⁢ ( 30375 ⁢ p 3 - 3 69603 ⁢ p 5 + 7 65765 ⁢ p 7 - 4 25425 ⁢ p 9 ) subscript 𝑈 3 𝑝 1 4 14720 30375 superscript 𝑝 3 3 69603 superscript 𝑝 5 7 65765 superscript 𝑝 7 4 25425 superscript 𝑝 9 {\displaystyle{\displaystyle U_{3}(p)=\tfrac{1}{4\;14720}\*(30375p^{3}-3\;6960% 3p^{5}+7\;65765p^{7}-4\;25425p^{9})}}
U_{3}(p) = \tfrac{1}{4\;14720}\*(30375p^{3}-3\;69603p^{5}+7\;65765p^{7}-4\;25425p^{9})		 

U[3](p) = (1)/(414720)*(30375*(p)^(3)- 369603*(p)^(5)+ 765765*(p)^(7)- 425425*(p)^(9))
 
Subscript[U, 3][p] == Divide[1,414720]*(30375*(p)^(3)- 369603*(p)^(5)+ 765765*(p)^(7)- 425425*(p)^(9))
 Skipped - no semantic math Skipped - no semantic math - -
10.41#Ex6 V 1 ⁢ ( p ) = 1 24 ⁢ ( - 9 ⁢ p + 7 ⁢ p 3 ) subscript 𝑉 1 𝑝 1 24 9 𝑝 7 superscript 𝑝 3 {\displaystyle{\displaystyle V_{1}(p)=\tfrac{1}{24}(-9p+7p^{3})}}
V_{1}(p) = \tfrac{1}{24}(-9p+7p^{3})		 

V[1](p) = (1)/(24)*(- 9*p + 7*(p)^(3))
 
Subscript[V, 1][p] == Divide[1,24]*(- 9*p + 7*(p)^(3))
 Skipped - no semantic math Skipped - no semantic math - -
10.41#Ex7 V 2 ⁢ ( p ) = 1 1152 ⁢ ( - 135 ⁢ p 2 + 594 ⁢ p 4 - 455 ⁢ p 6 ) subscript 𝑉 2 𝑝 1 1152 135 superscript 𝑝 2 594 superscript 𝑝 4 455 superscript 𝑝 6 {\displaystyle{\displaystyle V_{2}(p)=\tfrac{1}{1152}(-135p^{2}+594p^{4}-455p^% {6})}}
V_{2}(p) = \tfrac{1}{1152}(-135p^{2}+594p^{4}-455p^{6})		 

V[2](p) = (1)/(1152)*(- 135*(p)^(2)+ 594*(p)^(4)- 455*(p)^(6))
 
Subscript[V, 2][p] == Divide[1,1152]*(- 135*(p)^(2)+ 594*(p)^(4)- 455*(p)^(6))
 Skipped - no semantic math Skipped - no semantic math - -
10.41#Ex8 V 3 ⁢ ( p ) = 1 4 14720 ⁢ ( - 42525 ⁢ p 3 + 4 51737 ⁢ p 5 - 8 83575 ⁢ p 7 + 4 75475 ⁢ p 9 ) subscript 𝑉 3 𝑝 1 4 14720 42525 superscript 𝑝 3 4 51737 superscript 𝑝 5 8 83575 superscript 𝑝 7 4 75475 superscript 𝑝 9 {\displaystyle{\displaystyle V_{3}(p)=\tfrac{1}{4\;14720}\*(-42525p^{3}+4\;517% 37p^{5}-8\;83575p^{7}+4\;75475p^{9})}}
V_{3}(p) = \tfrac{1}{4\;14720}\*(-42525p^{3}+4\;51737p^{5}-8\;83575p^{7}+4\;75475p^{9})		 

V[3](p) = (1)/(414720)*(- 42525*(p)^(3)+ 451737*(p)^(5)- 883575*(p)^(7)+ 475475*(p)^(9))
 
Subscript[V, 3][p] == Divide[1,414720]*(- 42525*(p)^(3)+ 451737*(p)^(5)- 883575*(p)^(7)+ 475475*(p)^(9))
 Skipped - no semantic math Skipped - no semantic math - -
10.43.E4 ∫ 0 x I 0 ⁡ ( t ) - 1 t ⁢ d t = 1 2 ⁢ ∑ k = 1 ∞ ( - 1 ) k - 1 ⁢ ψ ⁡ ( k + 1 ) - ψ ⁡ ( 1 ) k ! ⁢ ( 1 2 ⁢ x ) k ⁢ I k ⁡ ( x ) superscript subscript 0 𝑥 modified-Bessel-first-kind 0 𝑡 1 𝑡 𝑡 1 2 superscript subscript 𝑘 1 superscript 1 𝑘 1 digamma 𝑘 1 digamma 1 𝑘 superscript 1 2 𝑥 𝑘 modified-Bessel-first-kind 𝑘 𝑥 {\displaystyle{\displaystyle\int_{0}^{x}\frac{I_{0}\left(t\right)-1}{t}\mathrm% {d}t=\frac{1}{2}\sum_{k=1}^{\infty}(-1)^{k-1}\frac{\psi\left(k+1\right)-\psi% \left(1\right)}{k!}(\tfrac{1}{2}x)^{k}I_{k}\left(x\right)}}
\int_{0}^{x}\frac{\modBesselI{0}@{t}-1}{t}\diff{t} = \frac{1}{2}\sum_{k=1}^{\infty}(-1)^{k-1}\frac{\digamma@{k+1}-\digamma@{1}}{k!}(\tfrac{1}{2}x)^{k}\modBesselI{k}@{x}		 ℜ ⁡ ( 0 + k + 1 ) > 0 , ℜ ⁡ ( k + k + 1 ) > 0 formulae-sequence 0 𝑘 1 0 𝑘 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(k+k+1)>0}} 
int((BesselI(0, t)- 1)/(t), t = 0..x) = (1)/(2)*sum((- 1)^(k - 1)*(Psi(k + 1)- Psi(1))/(factorial(k))*((1)/(2)*x)^(k)* BesselI(k, x), k = 1..infinity)
 
Integrate[Divide[BesselI[0, t]- 1,t], {t, 0, x}, GenerateConditions->None] == Divide[1,2]*Sum[(- 1)^(k - 1)*Divide[PolyGamma[k + 1]- PolyGamma[1],(k)!]*(Divide[1,2]*x)^(k)* BesselI[k, x], {k, 1, Infinity}, GenerateConditions->None]
 Failure Failure Successful [Tested: 3]
Failed [3 / 3]
Result: Plus[DirectedInfinity[-1], Times[-0.5, NSum[Times[Power[-1, Plus[-1, k]], Power[0.75, k], BesselI[k, 1.5], Power[Factorial[k], -1], Plus[EulerGamma, PolyGamma[0, Plus[1, k]]]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5]}

Result: Plus[DirectedInfinity[-1], Times[-0.5, NSum[Times[Power[-1, Plus[-1, k]], Power[0.25, k], BesselI[k, 0.5], Power[Factorial[k], -1], Plus[EulerGamma, PolyGamma[0, Plus[1, k]]]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 0.5]}

... skip entries to safe data
10.43.E4 1 2 ⁢ ∑ k = 1 ∞ ( - 1 ) k - 1 ⁢ ψ ⁡ ( k + 1 ) - ψ ⁡ ( 1 ) k ! ⁢ ( 1 2 ⁢ x ) k ⁢ I k ⁡ ( x ) = 2 x ⁢ ∑ k = 0 ∞ ( - 1 ) k ⁢ ( 2 ⁢ k + 3 ) ⁢ ( ψ ⁡ ( k + 2 ) - ψ ⁡ ( 1 ) ) ⁢ I 2 ⁢ k + 3 ⁡ ( x ) 1 2 superscript subscript 𝑘 1 superscript 1 𝑘 1 digamma 𝑘 1 digamma 1 𝑘 superscript 1 2 𝑥 𝑘 modified-Bessel-first-kind 𝑘 𝑥 2 𝑥 superscript subscript 𝑘 0 superscript 1 𝑘 2 𝑘 3 digamma 𝑘 2 digamma 1 modified-Bessel-first-kind 2 𝑘 3 𝑥 {\displaystyle{\displaystyle\frac{1}{2}\sum_{k=1}^{\infty}(-1)^{k-1}\frac{\psi% \left(k+1\right)-\psi\left(1\right)}{k!}(\tfrac{1}{2}x)^{k}I_{k}\left(x\right)% =\frac{2}{x}\sum_{k=0}^{\infty}(-1)^{k}(2k+3)(\psi\left(k+2\right)-\psi\left(1% \right))I_{2k+3}\left(x\right)}}
\frac{1}{2}\sum_{k=1}^{\infty}(-1)^{k-1}\frac{\digamma@{k+1}-\digamma@{1}}{k!}(\tfrac{1}{2}x)^{k}\modBesselI{k}@{x} = \frac{2}{x}\sum_{k=0}^{\infty}(-1)^{k}(2k+3)(\digamma@{k+2}-\digamma@{1})\modBesselI{2k+3}@{x}		 ℜ ⁡ ( 0 + k + 1 ) > 0 , ℜ ⁡ ( k + k + 1 ) > 0 , ℜ ⁡ ( ( 2 ⁢ k + 3 ) + k + 1 ) > 0 formulae-sequence 0 𝑘 1 0 formulae-sequence 𝑘 𝑘 1 0 2 𝑘 3 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(k+k+1)>0,\Re((2k+3)+k+1)>0}} 
(1)/(2)*sum((- 1)^(k - 1)*(Psi(k + 1)- Psi(1))/(factorial(k))*((1)/(2)*x)^(k)* BesselI(k, x), k = 1..infinity) = (2)/(x)*sum((- 1)^(k)*(2*k + 3)*(Psi(k + 2)- Psi(1))*BesselI(2*k + 3, x), k = 0..infinity)
 
Divide[1,2]*Sum[(- 1)^(k - 1)*Divide[PolyGamma[k + 1]- PolyGamma[1],(k)!]*(Divide[1,2]*x)^(k)* BesselI[k, x], {k, 1, Infinity}, GenerateConditions->None] == Divide[2,x]*Sum[(- 1)^(k)*(2*k + 3)*(PolyGamma[k + 2]- PolyGamma[1])*BesselI[2*k + 3, x], {k, 0, Infinity}, GenerateConditions->None]
 Failure Failure Successful [Tested: 3]
Failed [3 / 3]
Result: Plus[Times[0.5, NSum[Times[Power[-1, Plus[-1, k]], Power[0.75, k], BesselI[k, 1.5], Power[Factorial[k], -1], Plus[EulerGamma, PolyGamma[0, Plus[1, k]]]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], Times[-1.3333333333333333, NSum[Times[Power[-1, k], Plus[3, Times[2, k]], BesselI[Plus[3, Times[2, k]], 1.5], Plus[EulerGamma, PolyGamma[0, Plus[2, k]]]], {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5]}

Result: Plus[Times[0.5, NSum[Times[Power[-1, Plus[-1, k]], Power[0.25, k], BesselI[k, 0.5], Power[Factorial[k], -1], Plus[EulerGamma, PolyGamma[0, Plus[1, k]]]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], Times[-4.0, NSum[Times[Power[-1, k], Plus[3, Times[2, k]], BesselI[Plus[3, Times[2, k]], 0.5], Plus[EulerGamma, PolyGamma[0, Plus[2, k]]]], {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 0.5]}

... skip entries to safe data
10.43.E5 ∫ x ∞ K 0 ⁡ ( t ) t ⁢ d t = 1 2 ⁢ ( ln ⁡ ( 1 2 ⁢ x ) + γ ) 2 + π 2 24 - ∑ k = 1 ∞ ( ψ ⁡ ( k + 1 ) + 1 2 ⁢ k - ln ⁡ ( 1 2 ⁢ x ) ) ⁢ ( 1 2 ⁢ x ) 2 ⁢ k 2 ⁢ k ⁢ ( k ! ) 2 superscript subscript 𝑥 modified-Bessel-second-kind 0 𝑡 𝑡 𝑡 1 2 superscript 1 2 𝑥 2 superscript 𝜋 2 24 superscript subscript 𝑘 1 digamma 𝑘 1 1 2 𝑘 1 2 𝑥 superscript 1 2 𝑥 2 𝑘 2 𝑘 superscript 𝑘 2 {\displaystyle{\displaystyle\int_{x}^{\infty}\frac{K_{0}\left(t\right)}{t}% \mathrm{d}t=\frac{1}{2}\left(\ln\left(\tfrac{1}{2}x\right)+\gamma\right)^{2}+% \frac{\pi^{2}}{24}-\sum_{k=1}^{\infty}\left(\psi\left(k+1\right)+\frac{1}{2k}-% \ln\left(\tfrac{1}{2}x\right)\right)\frac{(\tfrac{1}{2}x)^{2k}}{2k(k!)^{2}}}}
\int_{x}^{\infty}\frac{\modBesselK{0}@{t}}{t}\diff{t} = \frac{1}{2}\left(\ln@{\tfrac{1}{2}x}+\EulerConstant\right)^{2}+\frac{\pi^{2}}{24}-\sum_{k=1}^{\infty}\left(\digamma@{k+1}+\frac{1}{2k}-\ln@{\tfrac{1}{2}x}\right)\frac{(\tfrac{1}{2}x)^{2k}}{2k(k!)^{2}}		 

int((BesselK(0, t))/(t), t = x..infinity) = (1)/(2)*(ln((1)/(2)*x)+ gamma)^(2)+((Pi)^(2))/(24)- sum((Psi(k + 1)+(1)/(2*k)- ln((1)/(2)*x))*(((1)/(2)*x)^(2*k))/(2*k*(factorial(k))^(2)), k = 1..infinity)
 
Integrate[Divide[BesselK[0, t],t], {t, x, Infinity}, GenerateConditions->None] == Divide[1,2]*(Log[Divide[1,2]*x]+ EulerGamma)^(2)+Divide[(Pi)^(2),24]- Sum[(PolyGamma[k + 1]+Divide[1,2*k]- Log[Divide[1,2]*x])*Divide[(Divide[1,2]*x)^(2*k),2*k*((k)!)^(2)], {k, 1, Infinity}, GenerateConditions->None]
 Failure Aborted Successful [Tested: 3] Skipped - Because timed out
10.43.E6 ∫ 0 x e - t ⁢ I n ⁡ ( t ) ⁢ d t = x ⁢ e - x ⁢ ( I 0 ⁡ ( x ) + I 1 ⁡ ( x ) ) + n ⁢ ( e - x ⁢ I 0 ⁡ ( x ) - 1 ) + 2 ⁢ e - x ⁢ ∑ k = 1 n - 1 ( n - k ) ⁢ I k ⁡ ( x ) superscript subscript 0 𝑥 superscript 𝑒 𝑡 modified-Bessel-first-kind 𝑛 𝑡 𝑡 𝑥 superscript 𝑒 𝑥 modified-Bessel-first-kind 0 𝑥 modified-Bessel-first-kind 1 𝑥 𝑛 superscript 𝑒 𝑥 modified-Bessel-first-kind 0 𝑥 1 2 superscript 𝑒 𝑥 superscript subscript 𝑘 1 𝑛 1 𝑛 𝑘 modified-Bessel-first-kind 𝑘 𝑥 {\displaystyle{\displaystyle\int_{0}^{x}e^{-t}I_{n}\left(t\right)\mathrm{d}t=% xe^{-x}(I_{0}\left(x\right)+I_{1}\left(x\right))+n(e^{-x}I_{0}\left(x\right)-1% )+2e^{-x}\sum_{k=1}^{n-1}(n-k)I_{k}\left(x\right)}}
\int_{0}^{x}e^{-t}\modBesselI{n}@{t}\diff{t} = xe^{-x}(\modBesselI{0}@{x}+\modBesselI{1}@{x})+n(e^{-x}\modBesselI{0}@{x}-1)+2e^{-x}\sum_{k=1}^{n-1}(n-k)\modBesselI{k}@{x}		 ℜ ⁡ ( n + k + 1 ) > 0 , ℜ ⁡ ( 0 + k + 1 ) > 0 , ℜ ⁡ ( 1 + k + 1 ) > 0 , ℜ ⁡ ( k + k + 1 ) > 0 formulae-sequence 𝑛 𝑘 1 0 formulae-sequence 0 𝑘 1 0 formulae-sequence 1 𝑘 1 0 𝑘 𝑘 1 0 {\displaystyle{\displaystyle\Re(n+k+1)>0,\Re(0+k+1)>0,\Re(1+k+1)>0,\Re(k+k+1)>% 0}} 
int(exp(- t)*BesselI(n, t), t = 0..x) = x*exp(- x)*(BesselI(0, x)+ BesselI(1, x))+ n*(exp(- x)*BesselI(0, x)- 1)+ 2*exp(- x)*sum((n - k)*BesselI(k, x), k = 1..n - 1)
 
Integrate[Exp[- t]*BesselI[n, t], {t, 0, x}, GenerateConditions->None] == x*Exp[- x]*(BesselI[0, x]+ BesselI[1, x])+ n*(Exp[- x]*BesselI[0, x]- 1)+ 2*Exp[- x]*Sum[(n - k)*BesselI[k, x], {k, 1, n - 1}, GenerateConditions->None]
 Failure Aborted Successful [Tested: 3]
Failed [2 / 3]
Result: Plus[1.0269197346695518, Times[-0.44626032029685964, Plus[-4.940169569318671, Times[3.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[1.5, []], Times[Plus[-2, Times[-2, ], Times[-1, 1.5]], [Plus[1, ]]], Times[Plus[2, Times[2, ], Times[-1, 1.5]], [Plus[2, ]]], Times[1.5, [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], BesselI[0, 1.5]], Equal[[2], Plus[BesselI[0, 1.5], BesselI[1, 1.5]]]}]][3.0]], Times[-1.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[2, ], 1.5, []], Times[-1, Plus[2, ], Plus[Times[2, ], 1.5], [Plus[1, ]]], Times[, Plus[4, Times[2, ], Times[-1, 1.5]], [Plus[2, ]]], Times[, 1.5, [Plus[3, ]]]], 0], Equal[[1], 0], Equal[[2], BesselI[1, 1.5]], Equal[[3], Plus[Times[2, Power[1.5, -1], Plus[Times[1.5, BesselI[0, 1.5]], Times[-2, BesselI[1, 1.5]]]], BesselI[1, 1.5]]]}]][3.0]]]]], {Rule[n, 3], Rule[x, 1.5]}

Result: Plus[0.6643873281588137, Times[-1.2130613194252668, Plus[-3.19045011222397, Times[3.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[0.5, []], Times[Plus[-2, Times[-2, ], Times[-1, 0.5]], [Plus[1, ]]], Times[Plus[2, Times[2, ], Times[-1, 0.5]], [Plus[2, ]]], Times[0.5, [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], BesselI[0, 0.5]], Equal[[2], Plus[BesselI[0, 0.5], BesselI[1, 0.5]]]}]][3.0]], Times[-1.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[2, ], 0.5, []], Times[-1, Plus[2, ], Plus[Times[2, ], 0.5], [Plus[1, ]]], Times[, Plus[4, Times[2, ], Times[-1, 0.5]], [Plus[2, ]]], Times[, 0.5, [Plus[3, ]]]], 0], Equal[[1], 0], Equal[[2], BesselI[1, 0.5]], Equal[[3], Plus[Times[2, Power[0.5, -1], Plus[Times[0.5, BesselI[0, 0.5]], Times[-2, BesselI[1, 0.5]]]], BesselI[1, 0.5]]]}]][3.0]]]]], {Rule[n, 3], Rule[x, 0.5]}

10.43.E7 ∫ 0 x e + t ⁢ t ν ⁢ I ν ⁡ ( t ) ⁢ d t = e + x ⁢ x ν + 1 2 ⁢ ν + 1 ⁢ ( I ν ⁡ ( x ) - I ν + 1 ⁡ ( x ) ) superscript subscript 0 𝑥 superscript 𝑒 𝑡 superscript 𝑡 𝜈 modified-Bessel-first-kind 𝜈 𝑡 𝑡 superscript 𝑒 𝑥 superscript 𝑥 𝜈 1 2 𝜈 1 modified-Bessel-first-kind 𝜈 𝑥 modified-Bessel-first-kind 𝜈 1 𝑥 {\displaystyle{\displaystyle\int_{0}^{x}e^{+t}t^{\nu}I_{\nu}\left(t\right)% \mathrm{d}t=\frac{e^{+x}x^{\nu+1}}{2\nu+1}(I_{\nu}\left(x\right)-I_{\nu+1}% \left(x\right))}}
\int_{0}^{x}e^{+ t}t^{\nu}\modBesselI{\nu}@{t}\diff{t} = \frac{e^{+ x}x^{\nu+1}}{2\nu+1}(\modBesselI{\nu}@{x}-\modBesselI{\nu+1}@{x})		 ℜ ⁡ ν > - 1 2 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( ν + 1 ) + k + 1 ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence 𝜈 𝑘 1 0 𝜈 1 𝑘 1 0 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+k+1)>0,\Re((\nu+1)+k+% 1)>0}} 
int(exp(+ t)*(t)^(nu)* BesselI(nu, t), t = 0..x) = (exp(+ x)*(x)^(nu + 1))/(2*nu + 1)*(BesselI(nu, x)- BesselI(nu + 1, x))
 
Integrate[Exp[+ t]*(t)^\[Nu]* BesselI[\[Nu], t], {t, 0, x}, GenerateConditions->None] == Divide[Exp[+ x]*(x)^(\[Nu]+ 1),2*\[Nu]+ 1]*(BesselI[\[Nu], x]- BesselI[\[Nu]+ 1, x])
 Failure Successful Successful [Tested: 15] Successful [Tested: 15]
10.43.E7 ∫ 0 x e - t ⁢ t ν ⁢ I ν ⁡ ( t ) ⁢ d t = e - x ⁢ x ν + 1 2 ⁢ ν + 1 ⁢ ( I ν ⁡ ( x ) + I ν + 1 ⁡ ( x ) ) superscript subscript 0 𝑥 superscript 𝑒 𝑡 superscript 𝑡 𝜈 modified-Bessel-first-kind 𝜈 𝑡 𝑡 superscript 𝑒 𝑥 superscript 𝑥 𝜈 1 2 𝜈 1 modified-Bessel-first-kind 𝜈 𝑥 modified-Bessel-first-kind 𝜈 1 𝑥 {\displaystyle{\displaystyle\int_{0}^{x}e^{-t}t^{\nu}I_{\nu}\left(t\right)% \mathrm{d}t=\frac{e^{-x}x^{\nu+1}}{2\nu+1}(I_{\nu}\left(x\right)+I_{\nu+1}% \left(x\right))}}
\int_{0}^{x}e^{- t}t^{\nu}\modBesselI{\nu}@{t}\diff{t} = \frac{e^{- x}x^{\nu+1}}{2\nu+1}(\modBesselI{\nu}@{x}+\modBesselI{\nu+1}@{x})		 ℜ ⁡ ν > - 1 2 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( ν + 1 ) + k + 1 ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence 𝜈 𝑘 1 0 𝜈 1 𝑘 1 0 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+k+1)>0,\Re((\nu+1)+k+% 1)>0}} 
int(exp(- t)*(t)^(nu)* BesselI(nu, t), t = 0..x) = (exp(- x)*(x)^(nu + 1))/(2*nu + 1)*(BesselI(nu, x)+ BesselI(nu + 1, x))
 
Integrate[Exp[- t]*(t)^\[Nu]* BesselI[\[Nu], t], {t, 0, x}, GenerateConditions->None] == Divide[Exp[- x]*(x)^(\[Nu]+ 1),2*\[Nu]+ 1]*(BesselI[\[Nu], x]+ BesselI[\[Nu]+ 1, x])
 Failure Successful Skipped - Because timed out Successful [Tested: 15]
10.43.E8 ∫ 0 x e + t ⁢ t - ν ⁢ I ν ⁡ ( t ) ⁢ d t = - e + x ⁢ x - ν + 1 2 ⁢ ν - 1 ⁢ ( I ν ⁡ ( x ) - I ν - 1 ⁡ ( x ) ) - 2 - ν + 1 ( 2 ⁢ ν - 1 ) ⁢ Γ ⁡ ( ν ) superscript subscript 0 𝑥 superscript 𝑒 𝑡 superscript 𝑡 𝜈 modified-Bessel-first-kind 𝜈 𝑡 𝑡 superscript 𝑒 𝑥 superscript 𝑥 𝜈 1 2 𝜈 1 modified-Bessel-first-kind 𝜈 𝑥 modified-Bessel-first-kind 𝜈 1 𝑥 superscript 2 𝜈 1 2 𝜈 1 Euler-Gamma 𝜈 {\displaystyle{\displaystyle\int_{0}^{x}e^{+t}t^{-\nu}I_{\nu}\left(t\right)% \mathrm{d}t=-\frac{e^{+x}x^{-\nu+1}}{2\nu-1}(I_{\nu}\left(x\right)-I_{\nu-1}% \left(x\right))-\frac{2^{-\nu+1}}{(2\nu-1)\Gamma\left(\nu\right)}}}
\int_{0}^{x}e^{+ t}t^{-\nu}\modBesselI{\nu}@{t}\diff{t} = -\frac{e^{+ x}x^{-\nu+1}}{2\nu-1}(\modBesselI{\nu}@{x}-\modBesselI{\nu-1}@{x})-\frac{2^{-\nu+1}}{(2\nu-1)\EulerGamma@{\nu}}		 ν ≠ 1 2 , ℜ ⁡ ( ν ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( ν - 1 ) + k + 1 ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence 𝜈 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 1 𝑘 1 0 {\displaystyle{\displaystyle\nu\neq\tfrac{1}{2},\Re(\nu)>0,\Re(\nu+k+1)>0,\Re(% (\nu-1)+k+1)>0}} 
int(exp(+ t)*(t)^(- nu)* BesselI(nu, t), t = 0..x) = -(exp(+ x)*(x)^(- nu + 1))/(2*nu - 1)*(BesselI(nu, x)- BesselI(nu - 1, x))-((2)^(- nu + 1))/((2*nu - 1)*GAMMA(nu))
 
Integrate[Exp[+ t]*(t)^(- \[Nu])* BesselI[\[Nu], t], {t, 0, x}, GenerateConditions->None] == -Divide[Exp[+ x]*(x)^(- \[Nu]+ 1),2*\[Nu]- 1]*(BesselI[\[Nu], x]- BesselI[\[Nu]- 1, x])-Divide[(2)^(- \[Nu]+ 1),(2*\[Nu]- 1)*Gamma[\[Nu]]]
 Failure Successful Manual Skip!
Failed [3 / 12]
Result: 0.39894228040143315
Test Values: {Rule[x, 1.5], Rule[ν, 1.5]}

Result: 0.39894228040143254
Test Values: {Rule[x, 0.5], Rule[ν, 1.5]}

... skip entries to safe data
10.43.E8 ∫ 0 x e - t ⁢ t - ν ⁢ I ν ⁡ ( t ) ⁢ d t = - e - x ⁢ x - ν + 1 2 ⁢ ν - 1 ⁢ ( I ν ⁡ ( x ) + I ν - 1 ⁡ ( x ) ) + 2 - ν + 1 ( 2 ⁢ ν - 1 ) ⁢ Γ ⁡ ( ν ) superscript subscript 0 𝑥 superscript 𝑒 𝑡 superscript 𝑡 𝜈 modified-Bessel-first-kind 𝜈 𝑡 𝑡 superscript 𝑒 𝑥 superscript 𝑥 𝜈 1 2 𝜈 1 modified-Bessel-first-kind 𝜈 𝑥 modified-Bessel-first-kind 𝜈 1 𝑥 superscript 2 𝜈 1 2 𝜈 1 Euler-Gamma 𝜈 {\displaystyle{\displaystyle\int_{0}^{x}e^{-t}t^{-\nu}I_{\nu}\left(t\right)% \mathrm{d}t=-\frac{e^{-x}x^{-\nu+1}}{2\nu-1}(I_{\nu}\left(x\right)+I_{\nu-1}% \left(x\right))+\frac{2^{-\nu+1}}{(2\nu-1)\Gamma\left(\nu\right)}}}
\int_{0}^{x}e^{- t}t^{-\nu}\modBesselI{\nu}@{t}\diff{t} = -\frac{e^{- x}x^{-\nu+1}}{2\nu-1}(\modBesselI{\nu}@{x}+\modBesselI{\nu-1}@{x})+\frac{2^{-\nu+1}}{(2\nu-1)\EulerGamma@{\nu}}		 ν ≠ 1 2 , ℜ ⁡ ( ν ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( ν - 1 ) + k + 1 ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence 𝜈 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 1 𝑘 1 0 {\displaystyle{\displaystyle\nu\neq\tfrac{1}{2},\Re(\nu)>0,\Re(\nu+k+1)>0,\Re(% (\nu-1)+k+1)>0}} 
int(exp(- t)*(t)^(- nu)* BesselI(nu, t), t = 0..x) = -(exp(- x)*(x)^(- nu + 1))/(2*nu - 1)*(BesselI(nu, x)+ BesselI(nu - 1, x))+((2)^(- nu + 1))/((2*nu - 1)*GAMMA(nu))
 
Integrate[Exp[- t]*(t)^(- \[Nu])* BesselI[\[Nu], t], {t, 0, x}, GenerateConditions->None] == -Divide[Exp[- x]*(x)^(- \[Nu]+ 1),2*\[Nu]- 1]*(BesselI[\[Nu], x]+ BesselI[\[Nu]- 1, x])+Divide[(2)^(- \[Nu]+ 1),(2*\[Nu]- 1)*Gamma[\[Nu]]]
 Failure Successful Manual Skip! Successful [Tested: 12]
10.43.E9 ∫ 0 x e + t ⁢ t ν ⁢ K ν ⁡ ( t ) ⁢ d t = e + x ⁢ x ν + 1 2 ⁢ ν + 1 ⁢ ( K ν ⁡ ( x ) + K ν + 1 ⁡ ( x ) ) - 2 ν ⁢ Γ ⁡ ( ν + 1 ) 2 ⁢ ν + 1 superscript subscript 0 𝑥 superscript 𝑒 𝑡 superscript 𝑡 𝜈 modified-Bessel-second-kind 𝜈 𝑡 𝑡 superscript 𝑒 𝑥 superscript 𝑥 𝜈 1 2 𝜈 1 modified-Bessel-second-kind 𝜈 𝑥 modified-Bessel-second-kind 𝜈 1 𝑥 superscript 2 𝜈 Euler-Gamma 𝜈 1 2 𝜈 1 {\displaystyle{\displaystyle\int_{0}^{x}e^{+t}t^{\nu}K_{\nu}\left(t\right)% \mathrm{d}t=\frac{e^{+x}x^{\nu+1}}{2\nu+1}(K_{\nu}\left(x\right)+K_{\nu+1}% \left(x\right))-\frac{2^{\nu}\Gamma\left(\nu+1\right)}{2\nu+1}}}
\int_{0}^{x}e^{+ t}t^{\nu}\modBesselK{\nu}@{t}\diff{t} = \frac{e^{+ x}x^{\nu+1}}{2\nu+1}(\modBesselK{\nu}@{x}+\modBesselK{\nu+1}@{x})-\frac{2^{\nu}\EulerGamma@{\nu+1}}{2\nu+1}		 ℜ ⁡ ν > - 1 2 , ℜ ⁡ ( ν + 1 ) > 0 formulae-sequence 𝜈 1 2 𝜈 1 0 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+1)>0}} 
int(exp(+ t)*(t)^(nu)* BesselK(nu, t), t = 0..x) = (exp(+ x)*(x)^(nu + 1))/(2*nu + 1)*(BesselK(nu, x)+ BesselK(nu + 1, x))-((2)^(nu)* GAMMA(nu + 1))/(2*nu + 1)
 
Integrate[Exp[+ t]*(t)^\[Nu]* BesselK[\[Nu], t], {t, 0, x}, GenerateConditions->None] == Divide[Exp[+ x]*(x)^(\[Nu]+ 1),2*\[Nu]+ 1]*(BesselK[\[Nu], x]+ BesselK[\[Nu]+ 1, x])-Divide[(2)^\[Nu]* Gamma[\[Nu]+ 1],2*\[Nu]+ 1]
 Failure Aborted Manual Skip!
Failed [9 / 15]
Result: DirectedInfinity[]
Test Values: {Rule[x, 1.5], Rule[ν, 1.5]}

Result: DirectedInfinity[]
Test Values: {Rule[x, 1.5], Rule[ν, 0.5]}

... skip entries to safe data
10.43.E9 ∫ 0 x e - t ⁢ t ν ⁢ K ν ⁡ ( t ) ⁢ d t = e - x ⁢ x ν + 1 2 ⁢ ν + 1 ⁢ ( K ν ⁡ ( x ) - K ν + 1 ⁡ ( x ) ) + 2 ν ⁢ Γ ⁡ ( ν + 1 ) 2 ⁢ ν + 1 superscript subscript 0 𝑥 superscript 𝑒 𝑡 superscript 𝑡 𝜈 modified-Bessel-second-kind 𝜈 𝑡 𝑡 superscript 𝑒 𝑥 superscript 𝑥 𝜈 1 2 𝜈 1 modified-Bessel-second-kind 𝜈 𝑥 modified-Bessel-second-kind 𝜈 1 𝑥 superscript 2 𝜈 Euler-Gamma 𝜈 1 2 𝜈 1 {\displaystyle{\displaystyle\int_{0}^{x}e^{-t}t^{\nu}K_{\nu}\left(t\right)% \mathrm{d}t=\frac{e^{-x}x^{\nu+1}}{2\nu+1}(K_{\nu}\left(x\right)-K_{\nu+1}% \left(x\right))+\frac{2^{\nu}\Gamma\left(\nu+1\right)}{2\nu+1}}}
\int_{0}^{x}e^{- t}t^{\nu}\modBesselK{\nu}@{t}\diff{t} = \frac{e^{- x}x^{\nu+1}}{2\nu+1}(\modBesselK{\nu}@{x}-\modBesselK{\nu+1}@{x})+\frac{2^{\nu}\EulerGamma@{\nu+1}}{2\nu+1}		 ℜ ⁡ ν > - 1 2 , ℜ ⁡ ( ν + 1 ) > 0 formulae-sequence 𝜈 1 2 𝜈 1 0 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+1)>0}} 
int(exp(- t)*(t)^(nu)* BesselK(nu, t), t = 0..x) = (exp(- x)*(x)^(nu + 1))/(2*nu + 1)*(BesselK(nu, x)- BesselK(nu + 1, x))+((2)^(nu)* GAMMA(nu + 1))/(2*nu + 1)
 
Integrate[Exp[- t]*(t)^\[Nu]* BesselK[\[Nu], t], {t, 0, x}, GenerateConditions->None] == Divide[Exp[- x]*(x)^(\[Nu]+ 1),2*\[Nu]+ 1]*(BesselK[\[Nu], x]- BesselK[\[Nu]+ 1, x])+Divide[(2)^\[Nu]* Gamma[\[Nu]+ 1],2*\[Nu]+ 1]
 Failure Successful Manual Skip!
Failed [3 / 15]
Result: DirectedInfinity[]
Test Values: {Rule[x, 1.5], Rule[ν, 2]}

Result: DirectedInfinity[]
Test Values: {Rule[x, 0.5], Rule[ν, 2]}

... skip entries to safe data
10.43.E10 ∫ x ∞ e t ⁢ t - ν ⁢ K ν ⁡ ( t ) ⁢ d t = e x ⁢ x - ν + 1 2 ⁢ ν - 1 ⁢ ( K ν ⁡ ( x ) + K ν - 1 ⁡ ( x ) ) superscript subscript 𝑥 superscript 𝑒 𝑡 superscript 𝑡 𝜈 modified-Bessel-second-kind 𝜈 𝑡 𝑡 superscript 𝑒 𝑥 superscript 𝑥 𝜈 1 2 𝜈 1 modified-Bessel-second-kind 𝜈 𝑥 modified-Bessel-second-kind 𝜈 1 𝑥 {\displaystyle{\displaystyle\int_{x}^{\infty}e^{t}t^{-\nu}K_{\nu}\left(t\right% )\mathrm{d}t=\frac{e^{x}x^{-\nu+1}}{2\nu-1}(K_{\nu}\left(x\right)+K_{\nu-1}% \left(x\right))}}
\int_{x}^{\infty}e^{t}t^{-\nu}\modBesselK{\nu}@{t}\diff{t} = \frac{e^{x}x^{-\nu+1}}{2\nu-1}(\modBesselK{\nu}@{x}+\modBesselK{\nu-1}@{x})		 ℜ ⁡ ν > 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\nu>\tfrac{1}{2}}} 
int(exp(t)*(t)^(- nu)* BesselK(nu, t), t = x..infinity) = (exp(x)*(x)^(- nu + 1))/(2*nu - 1)*(BesselK(nu, x)+ BesselK(nu - 1, x))
 
Integrate[Exp[t]*(t)^(- \[Nu])* BesselK[\[Nu], t], {t, x, Infinity}, GenerateConditions->None] == Divide[Exp[x]*(x)^(- \[Nu]+ 1),2*\[Nu]- 1]*(BesselK[\[Nu], x]+ BesselK[\[Nu]- 1, x])
 Failure Successful Manual Skip!
Failed [3 / 9]
Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[ν, 2]}

Result: DirectedInfinity[]
Test Values: {Rule[x, 0.5], Rule[ν, 2]}

... skip entries to safe data
10.43.E18 ∫ 0 ∞ K ν ⁡ ( t ) ⁢ d t = 1 2 ⁢ π ⁢ sec ⁡ ( 1 2 ⁢ π ⁢ ν ) superscript subscript 0 modified-Bessel-second-kind 𝜈 𝑡 𝑡 1 2 𝜋 1 2 𝜋 𝜈 {\displaystyle{\displaystyle\int_{0}^{\infty}K_{\nu}\left(t\right)\mathrm{d}t=% \tfrac{1}{2}\pi\sec\left(\tfrac{1}{2}\pi\nu\right)}}
\int_{0}^{\infty}\modBesselK{\nu}@{t}\diff{t} = \tfrac{1}{2}\pi\sec@{\tfrac{1}{2}\pi\nu}		 | ℜ ⁡ ν | < 1 𝜈 1 {\displaystyle{\displaystyle|\Re\nu|<1}} 
int(BesselK(nu, t), t = 0..infinity) = (1)/(2)*Pi*sec((1)/(2)*Pi*nu)
 
Integrate[BesselK[\[Nu], t], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,2]*Pi*Sec[Divide[1,2]*Pi*\[Nu]]
 Successful Successful - Successful [Tested: 6]
10.43.E19 ∫ 0 ∞ t μ - 1 ⁢ K ν ⁡ ( t ) ⁢ d t = 2 μ - 2 ⁢ Γ ⁡ ( 1 2 ⁢ μ - 1 2 ⁢ ν ) ⁢ Γ ⁡ ( 1 2 ⁢ μ + 1 2 ⁢ ν ) superscript subscript 0 superscript 𝑡 𝜇 1 modified-Bessel-second-kind 𝜈 𝑡 𝑡 superscript 2 𝜇 2 Euler-Gamma 1 2 𝜇 1 2 𝜈 Euler-Gamma 1 2 𝜇 1 2 𝜈 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu-1}K_{\nu}\left(t\right)% \mathrm{d}t=2^{\mu-2}\Gamma\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu\right)\Gamma% \left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu\right)}}
\int_{0}^{\infty}t^{\mu-1}\modBesselK{\nu}@{t}\diff{t} = 2^{\mu-2}\EulerGamma@{\tfrac{1}{2}\mu-\tfrac{1}{2}\nu}\EulerGamma@{\tfrac{1}{2}\mu+\tfrac{1}{2}\nu}		 | ℜ ⁡ ν | < ℜ ⁡ μ , ℜ ⁡ ( 1 2 ⁢ μ - 1 2 ⁢ ν ) > 0 , ℜ ⁡ ( 1 2 ⁢ μ + 1 2 ⁢ ν ) > 0 formulae-sequence 𝜈 𝜇 formulae-sequence 1 2 𝜇 1 2 𝜈 0 1 2 𝜇 1 2 𝜈 0 {\displaystyle{\displaystyle|\Re\nu|<\Re\mu,\Re(\tfrac{1}{2}\mu-\tfrac{1}{2}% \nu)>0,\Re(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu)>0}} 
int((t)^(mu - 1)* BesselK(nu, t), t = 0..infinity) = (2)^(mu - 2)* GAMMA((1)/(2)*mu -(1)/(2)*nu)*GAMMA((1)/(2)*mu +(1)/(2)*nu)
 
Integrate[(t)^(\[Mu]- 1)* BesselK[\[Nu], t], {t, 0, Infinity}, GenerateConditions->None] == (2)^(\[Mu]- 2)* Gamma[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]]*Gamma[Divide[1,2]*\[Mu]+Divide[1,2]*\[Nu]]
 Successful Successful - Successful [Tested: 18]
10.43.E20 ∫ 0 ∞ cos ⁡ ( a ⁢ t ) ⁢ K 0 ⁡ ( t ) ⁢ d t = π 2 ⁢ ( 1 + a 2 ) 1 2 superscript subscript 0 𝑎 𝑡 modified-Bessel-second-kind 0 𝑡 𝑡 𝜋 2 superscript 1 superscript 𝑎 2 1 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\cos\left(at\right)K_{0}\left(t% \right)\mathrm{d}t=\frac{\pi}{2(1+a^{2})^{\frac{1}{2}}}}}
\int_{0}^{\infty}\cos@{at}\modBesselK{0}@{t}\diff{t} = \frac{\pi}{2(1+a^{2})^{\frac{1}{2}}}		 | ℑ ⁡ a | < 1 𝑎 1 {\displaystyle{\displaystyle|\Im a|<1}} 
int(cos(a*t)*BesselK(0, t), t = 0..infinity) = (Pi)/(2*(1 + (a)^(2))^((1)/(2)))
 
Integrate[Cos[a*t]*BesselK[0, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Pi,2*(1 + (a)^(2))^(Divide[1,2])]
 Successful Aborted - Successful [Tested: 6]
10.43.E21 ∫ 0 ∞ sin ⁡ ( a ⁢ t ) ⁢ K 0 ⁡ ( t ) ⁢ d t = arcsinh ⁡ a ( 1 + a 2 ) 1 2 superscript subscript 0 𝑎 𝑡 modified-Bessel-second-kind 0 𝑡 𝑡 hyperbolic-inverse-sine 𝑎 superscript 1 superscript 𝑎 2 1 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\sin\left(at\right)K_{0}\left(t% \right)\mathrm{d}t=\frac{\operatorname{arcsinh}a}{(1+a^{2})^{\frac{1}{2}}}}}
\int_{0}^{\infty}\sin@{at}\modBesselK{0}@{t}\diff{t} = \frac{\asinh@@{a}}{(1+a^{2})^{\frac{1}{2}}}		 | ℑ ⁡ a | < 1 𝑎 1 {\displaystyle{\displaystyle|\Im a|<1}} 
int(sin(a*t)*BesselK(0, t), t = 0..infinity) = (arcsinh(a))/((1 + (a)^(2))^((1)/(2)))
 
Integrate[Sin[a*t]*BesselK[0, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[ArcSinh[a],(1 + (a)^(2))^(Divide[1,2])]
 Failure Successful Successful [Tested: 0] Successful [Tested: 6]
10.43.E23 ∫ 0 ∞ t ν + 1 ⁢ I ν ⁡ ( b ⁢ t ) ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ d t = b ν ( 2 ⁢ p 2 ) ν + 1 ⁢ exp ⁡ ( b 2 4 ⁢ p 2 ) superscript subscript 0 superscript 𝑡 𝜈 1 modified-Bessel-first-kind 𝜈 𝑏 𝑡 superscript 𝑝 2 superscript 𝑡 2 𝑡 superscript 𝑏 𝜈 superscript 2 superscript 𝑝 2 𝜈 1 superscript 𝑏 2 4 superscript 𝑝 2 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\nu+1}I_{\nu}\left(bt\right)% \exp\left(-p^{2}t^{2}\right)\mathrm{d}t=\frac{b^{\nu}}{(2p^{2})^{\nu+1}}\exp% \left(\frac{b^{2}}{4p^{2}}\right)}}
\int_{0}^{\infty}t^{\nu+1}\modBesselI{\nu}@{bt}\exp@{-p^{2}t^{2}}\diff{t} = \frac{b^{\nu}}{(2p^{2})^{\nu+1}}\exp@{\frac{b^{2}}{4p^{2}}}		 ℜ ⁡ ν > - 1 , ℜ ⁡ ( p 2 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 1 formulae-sequence superscript 𝑝 2 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\nu>-1,\Re\left(p^{2}\right)>0,\Re(\nu+k+1)>0}} 
int((t)^(nu + 1)* BesselI(nu, b*t)*exp(- (p)^(2)* (t)^(2)), t = 0..infinity) = ((b)^(nu))/((2*(p)^(2))^(nu + 1))*exp(((b)^(2))/(4*(p)^(2)))
 
Integrate[(t)^(\[Nu]+ 1)* BesselI[\[Nu], b*t]*Exp[- (p)^(2)* (t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Divide[(b)^\[Nu],(2*(p)^(2))^(\[Nu]+ 1)]*Exp[Divide[(b)^(2),4*(p)^(2)]]
 Error Aborted - Skip - No test values generated
10.43.E24 ∫ 0 ∞ I ν ⁡ ( b ⁢ t ) ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ d t = π 2 ⁢ p ⁢ exp ⁡ ( b 2 8 ⁢ p 2 ) ⁢ I 1 2 ⁢ ν ⁡ ( b 2 8 ⁢ p 2 ) superscript subscript 0 modified-Bessel-first-kind 𝜈 𝑏 𝑡 superscript 𝑝 2 superscript 𝑡 2 𝑡 𝜋 2 𝑝 superscript 𝑏 2 8 superscript 𝑝 2 modified-Bessel-first-kind 1 2 𝜈 superscript 𝑏 2 8 superscript 𝑝 2 {\displaystyle{\displaystyle\int_{0}^{\infty}I_{\nu}\left(bt\right)\exp\left(-% p^{2}t^{2}\right)\mathrm{d}t=\frac{\sqrt{\pi}}{2p}\exp\left(\frac{b^{2}}{8p^{2% }}\right)I_{\frac{1}{2}\nu}\left(\frac{b^{2}}{8p^{2}}\right)}}
\int_{0}^{\infty}\modBesselI{\nu}@{bt}\exp@{-p^{2}t^{2}}\diff{t} = \frac{\sqrt{\pi}}{2p}\exp@{\frac{b^{2}}{8p^{2}}}\modBesselI{\frac{1}{2}\nu}@{\frac{b^{2}}{8p^{2}}}		 ℜ ⁡ ν > - 1 , ℜ ⁡ ( p 2 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( 1 2 ⁢ ν ) + k + 1 ) > 0 formulae-sequence 𝜈 1 formulae-sequence superscript 𝑝 2 0 formulae-sequence 𝜈 𝑘 1 0 1 2 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\nu>-1,\Re\left(p^{2}\right)>0,\Re(\nu+k+1)>0,% \Re((\frac{1}{2}\nu)+k+1)>0}} 
int(BesselI(nu, b*t)*exp(- (p)^(2)* (t)^(2)), t = 0..infinity) = (sqrt(Pi))/(2*p)*exp(((b)^(2))/(8*(p)^(2)))*BesselI((1)/(2)*nu, ((b)^(2))/(8*(p)^(2)))
 
Integrate[BesselI[\[Nu], b*t]*Exp[- (p)^(2)* (t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Divide[Sqrt[Pi],2*p]*Exp[Divide[(b)^(2),8*(p)^(2)]]*BesselI[Divide[1,2]*\[Nu], Divide[(b)^(2),8*(p)^(2)]]
 Failure Aborted
Failed [228 / 300]
Result: -.7585567167+3.675115279*I
Test Values: {b = -3/2, nu = 1/2*3^(1/2)+1/2*I, p = 1/2*3^(1/2)+1/2*I}

Result: -.9489546609+2.381017603*I
Test Values: {b = -3/2, nu = 1/2*3^(1/2)+1/2*I, p = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [152 / 300]
Result: Complex[-0.19039794459564638, -1.294097675814569]
Test Values: {Rule[b, -1.5], Rule[p, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[2.992047945390181, -4.249025046528451]
Test Values: {Rule[b, -1.5], Rule[p, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.43.E25 ∫ 0 ∞ K ν ⁡ ( b ⁢ t ) ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ d t = π 4 ⁢ p ⁢ sec ⁡ ( 1 2 ⁢ π ⁢ ν ) ⁢ exp ⁡ ( b 2 8 ⁢ p 2 ) ⁢ K 1 2 ⁢ ν ⁡ ( b 2 8 ⁢ p 2 ) superscript subscript 0 modified-Bessel-second-kind 𝜈 𝑏 𝑡 superscript 𝑝 2 superscript 𝑡 2 𝑡 𝜋 4 𝑝 1 2 𝜋 𝜈 superscript 𝑏 2 8 superscript 𝑝 2 modified-Bessel-second-kind 1 2 𝜈 superscript 𝑏 2 8 superscript 𝑝 2 {\displaystyle{\displaystyle\int_{0}^{\infty}K_{\nu}\left(bt\right)\exp\left(-% p^{2}t^{2}\right)\mathrm{d}t=\frac{\sqrt{\pi}}{4p}\sec\left(\tfrac{1}{2}\pi\nu% \right)\exp\left(\frac{b^{2}}{8p^{2}}\right)K_{\frac{1}{2}\nu}\left(\frac{b^{2% }}{8p^{2}}\right)}}
\int_{0}^{\infty}\modBesselK{\nu}@{bt}\exp@{-p^{2}t^{2}}\diff{t} = \frac{\sqrt{\pi}}{4p}\sec@{\tfrac{1}{2}\pi\nu}\exp@{\frac{b^{2}}{8p^{2}}}\modBesselK{\frac{1}{2}\nu}@{\frac{b^{2}}{8p^{2}}}		 | ℜ ⁡ ν | < 1 , ℜ ⁡ ( p 2 ) > 0 formulae-sequence 𝜈 1 superscript 𝑝 2 0 {\displaystyle{\displaystyle|\Re\nu|<1,\Re\left(p^{2}\right)>0}} 
int(BesselK(nu, b*t)*exp(- (p)^(2)* (t)^(2)), t = 0..infinity) = (sqrt(Pi))/(4*p)*sec((1)/(2)*Pi*nu)*exp(((b)^(2))/(8*(p)^(2)))*BesselK((1)/(2)*nu, ((b)^(2))/(8*(p)^(2)))
 
Integrate[BesselK[\[Nu], b*t]*Exp[- (p)^(2)* (t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Divide[Sqrt[Pi],4*p]*Sec[Divide[1,2]*Pi*\[Nu]]*Exp[Divide[(b)^(2),8*(p)^(2)]]*BesselK[Divide[1,2]*\[Nu], Divide[(b)^(2),8*(p)^(2)]]
 Failure Aborted
Failed [144 / 288]
Result: -.4056916296-1.844454275*I
Test Values: {b = -3/2, nu = 1/2*3^(1/2)+1/2*I, p = 1/2*3^(1/2)+1/2*I}

Result: -.2830456904e-1-1.996429597*I
Test Values: {b = -3/2, nu = 1/2*3^(1/2)+1/2*I, p = 3/2}

... skip entries to safe data
Failed [144 / 288]
Result: Complex[0.40569163152223653, 1.8444542715605226]
Test Values: {Rule[b, -1.5], Rule[p, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[0.4232355421098407, -0.8203643961026106]
Test Values: {Rule[b, -1.5], Rule[p, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.43.E26 ∫ 0 ∞ K μ ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( b ⁢ t ) t λ ⁢ d t = b ν ⁢ Γ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ λ + 1 2 ⁢ μ + 1 2 ) ⁢ Γ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ λ - 1 2 ⁢ μ + 1 2 ) 2 λ + 1 ⁢ a ν - λ + 1 ⁢ 𝐅 ⁡ ( ν - λ + μ + 1 2 , ν - λ - μ + 1 2 ; ν + 1 ; - b 2 a 2 ) superscript subscript 0 modified-Bessel-second-kind 𝜇 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 superscript 𝑡 𝜆 𝑡 superscript 𝑏 𝜈 Euler-Gamma 1 2 𝜈 1 2 𝜆 1 2 𝜇 1 2 Euler-Gamma 1 2 𝜈 1 2 𝜆 1 2 𝜇 1 2 superscript 2 𝜆 1 superscript 𝑎 𝜈 𝜆 1 scaled-hypergeometric-bold-F 𝜈 𝜆 𝜇 1 2 𝜈 𝜆 𝜇 1 2 𝜈 1 superscript 𝑏 2 superscript 𝑎 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{K_{\mu}\left(at\right)J_{% \nu}\left(bt\right)}{t^{\lambda}}\mathrm{d}t=\frac{b^{\nu}\Gamma\left(\frac{1}% {2}\nu-\frac{1}{2}\lambda+\frac{1}{2}\mu+\frac{1}{2}\right)\Gamma\left(\frac{1% }{2}\nu-\frac{1}{2}\lambda-\frac{1}{2}\mu+\frac{1}{2}\right)}{2^{\lambda+1}a^{% \nu-\lambda+1}}\*\mathbf{F}\left(\frac{\nu-\lambda+\mu+1}{2},\frac{\nu-\lambda% -\mu+1}{2};\nu+1;-\frac{b^{2}}{a^{2}}\right)}}
\int_{0}^{\infty}\frac{\modBesselK{\mu}@{at}\BesselJ{\nu}@{bt}}{t^{\lambda}}\diff{t} = \frac{b^{\nu}\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\lambda+\frac{1}{2}\mu+\frac{1}{2}}\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\lambda-\frac{1}{2}\mu+\frac{1}{2}}}{2^{\lambda+1}a^{\nu-\lambda+1}}\*\hyperOlverF@{\frac{\nu-\lambda+\mu+1}{2}}{\frac{\nu-\lambda-\mu+1}{2}}{\nu+1}{-\frac{b^{2}}{a^{2}}}		 ℜ ⁡ ( ν + 1 - λ ) > | ℜ ⁡ μ | , ℜ ⁡ a > | ℑ ⁡ b | , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ λ + 1 2 ⁢ μ + 1 2 ) > 0 , ℜ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ λ - 1 2 ⁢ μ + 1 2 ) > 0 formulae-sequence 𝜈 1 𝜆 𝜇 formulae-sequence 𝑎 𝑏 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 1 2 𝜈 1 2 𝜆 1 2 𝜇 1 2 0 1 2 𝜈 1 2 𝜆 1 2 𝜇 1 2 0 {\displaystyle{\displaystyle\Re\left(\nu+1-\lambda\right)>|\Re\mu|,\Re a>|\Im b% |,\Re(\nu+k+1)>0,\Re(\frac{1}{2}\nu-\frac{1}{2}\lambda+\frac{1}{2}\mu+\frac{1}% {2})>0,\Re(\frac{1}{2}\nu-\frac{1}{2}\lambda-\frac{1}{2}\mu+\frac{1}{2})>0}} 
int((BesselK(mu, a*t)*BesselJ(nu, b*t))/((t)^(lambda)), t = 0..infinity) = ((b)^(nu)* GAMMA((1)/(2)*nu -(1)/(2)*lambda +(1)/(2)*mu +(1)/(2))*GAMMA((1)/(2)*nu -(1)/(2)*lambda -(1)/(2)*mu +(1)/(2)))/((2)^(lambda + 1)* (a)^(nu - lambda + 1))* hypergeom([(nu - lambda + mu + 1)/(2), (nu - lambda - mu + 1)/(2)], [nu + 1], -((b)^(2))/((a)^(2)))/GAMMA(nu + 1)
 
Integrate[Divide[BesselK[\[Mu], a*t]*BesselJ[\[Nu], b*t],(t)^\[Lambda]], {t, 0, Infinity}, GenerateConditions->None] == Divide[(b)^\[Nu]* Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Lambda]+Divide[1,2]*\[Mu]+Divide[1,2]]*Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Lambda]-Divide[1,2]*\[Mu]+Divide[1,2]],(2)^(\[Lambda]+ 1)* (a)^(\[Nu]- \[Lambda]+ 1)]* Hypergeometric2F1Regularized[Divide[\[Nu]- \[Lambda]+ \[Mu]+ 1,2], Divide[\[Nu]- \[Lambda]- \[Mu]+ 1,2], \[Nu]+ 1, -Divide[(b)^(2),(a)^(2)]]
 Error Aborted - Skip - No test values generated
10.43.E27 ∫ 0 ∞ t μ + ν + 1 ⁢ K μ ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( b ⁢ t ) ⁢ d t = ( 2 ⁢ a ) μ ⁢ ( 2 ⁢ b ) ν ⁢ Γ ⁡ ( μ + ν + 1 ) ( a 2 + b 2 ) μ + ν + 1 superscript subscript 0 superscript 𝑡 𝜇 𝜈 1 modified-Bessel-second-kind 𝜇 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 𝑡 superscript 2 𝑎 𝜇 superscript 2 𝑏 𝜈 Euler-Gamma 𝜇 𝜈 1 superscript superscript 𝑎 2 superscript 𝑏 2 𝜇 𝜈 1 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu+\nu+1}K_{\mu}\left(at% \right)J_{\nu}\left(bt\right)\mathrm{d}t=\frac{(2a)^{\mu}(2b)^{\nu}\Gamma\left% (\mu+\nu+1\right)}{(a^{2}+b^{2})^{\mu+\nu+1}}}}
\int_{0}^{\infty}t^{\mu+\nu+1}\modBesselK{\mu}@{at}\BesselJ{\nu}@{bt}\diff{t} = \frac{(2a)^{\mu}(2b)^{\nu}\EulerGamma@{\mu+\nu+1}}{(a^{2}+b^{2})^{\mu+\nu+1}}		 ℜ ⁡ ( ν + 1 ) > | ℜ ⁡ μ | , ℜ ⁡ a > | ℑ ⁡ b | , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( μ + ν + 1 ) > 0 formulae-sequence 𝜈 1 𝜇 formulae-sequence 𝑎 𝑏 formulae-sequence 𝜈 𝑘 1 0 𝜇 𝜈 1 0 {\displaystyle{\displaystyle\Re\left(\nu+1\right)>|\Re\mu|,\Re a>|\Im b|,\Re(% \nu+k+1)>0,\Re(\mu+\nu+1)>0}} 
int((t)^(mu + nu + 1)* BesselK(mu, a*t)*BesselJ(nu, b*t), t = 0..infinity) = ((2*a)^(mu)*(2*b)^(nu)* GAMMA(mu + nu + 1))/(((a)^(2)+ (b)^(2))^(mu + nu + 1))
 
Integrate[(t)^(\[Mu]+ \[Nu]+ 1)* BesselK[\[Mu], a*t]*BesselJ[\[Nu], b*t], {t, 0, Infinity}, GenerateConditions->None] == Divide[(2*a)^\[Mu]*(2*b)^\[Nu]* Gamma[\[Mu]+ \[Nu]+ 1],((a)^(2)+ (b)^(2))^(\[Mu]+ \[Nu]+ 1)]
 Error Aborted - Skip - No test values generated
10.43.E28 ∫ 0 ∞ t ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ I ν ⁡ ( a ⁢ t ) ⁢ I ν ⁡ ( b ⁢ t ) ⁢ d t = 1 2 ⁢ p 2 ⁢ exp ⁡ ( a 2 + b 2 4 ⁢ p 2 ) ⁢ I ν ⁡ ( a ⁢ b 2 ⁢ p 2 ) superscript subscript 0 𝑡 superscript 𝑝 2 superscript 𝑡 2 modified-Bessel-first-kind 𝜈 𝑎 𝑡 modified-Bessel-first-kind 𝜈 𝑏 𝑡 𝑡 1 2 superscript 𝑝 2 superscript 𝑎 2 superscript 𝑏 2 4 superscript 𝑝 2 modified-Bessel-first-kind 𝜈 𝑎 𝑏 2 superscript 𝑝 2 {\displaystyle{\displaystyle\int_{0}^{\infty}t\exp\left(-p^{2}t^{2}\right)I_{% \nu}\left(at\right)I_{\nu}\left(bt\right)\mathrm{d}t=\frac{1}{2p^{2}}\exp\left% (\frac{a^{2}+b^{2}}{4p^{2}}\right)I_{\nu}\left(\frac{ab}{2p^{2}}\right)}}
\int_{0}^{\infty}t\exp@{-p^{2}t^{2}}\modBesselI{\nu}@{at}\modBesselI{\nu}@{bt}\diff{t} = \frac{1}{2p^{2}}\exp@{\frac{a^{2}+b^{2}}{4p^{2}}}\modBesselI{\nu}@{\frac{ab}{2p^{2}}}		 ℜ ⁡ ν > - 1 , ℜ ⁡ ( p 2 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 1 formulae-sequence superscript 𝑝 2 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\nu>-1,\Re\left(p^{2}\right)>0,\Re(\nu+k+1)>0}} 
int(t*exp(- (p)^(2)* (t)^(2))*BesselI(nu, a*t)*BesselI(nu, b*t), t = 0..infinity) = (1)/(2*(p)^(2))*exp(((a)^(2)+ (b)^(2))/(4*(p)^(2)))*BesselI(nu, (a*b)/(2*(p)^(2)))
 
Integrate[t*Exp[- (p)^(2)* (t)^(2)]*BesselI[\[Nu], a*t]*BesselI[\[Nu], b*t], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,2*(p)^(2)]*Exp[Divide[(a)^(2)+ (b)^(2),4*(p)^(2)]]*BesselI[\[Nu], Divide[a*b,2*(p)^(2)]]
 Error Aborted - Skipped - Because timed out
10.43.E29 ∫ 0 ∞ t ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ I 0 ⁡ ( a ⁢ t ) ⁢ K 0 ⁡ ( a ⁢ t ) ⁢ d t = 1 4 ⁢ p 2 ⁢ exp ⁡ ( a 2 2 ⁢ p 2 ) ⁢ K 0 ⁡ ( a 2 2 ⁢ p 2 ) superscript subscript 0 𝑡 superscript 𝑝 2 superscript 𝑡 2 modified-Bessel-first-kind 0 𝑎 𝑡 modified-Bessel-second-kind 0 𝑎 𝑡 𝑡 1 4 superscript 𝑝 2 superscript 𝑎 2 2 superscript 𝑝 2 modified-Bessel-second-kind 0 superscript 𝑎 2 2 superscript 𝑝 2 {\displaystyle{\displaystyle\int_{0}^{\infty}t\exp\left(-p^{2}t^{2}\right)I_{0% }\left(at\right)K_{0}\left(at\right)\mathrm{d}t=\frac{1}{4p^{2}}\exp\left(% \frac{a^{2}}{2p^{2}}\right)K_{0}\left(\frac{a^{2}}{2p^{2}}\right)}}
\int_{0}^{\infty}t\exp@{-p^{2}t^{2}}\modBesselI{0}@{at}\modBesselK{0}@{at}\diff{t} = \frac{1}{4p^{2}}\exp@{\frac{a^{2}}{2p^{2}}}\modBesselK{0}@{\frac{a^{2}}{2p^{2}}}		 ℜ ⁡ ( p 2 ) > 0 , ℜ ⁡ ( 0 + k + 1 ) > 0 formulae-sequence superscript 𝑝 2 0 0 𝑘 1 0 {\displaystyle{\displaystyle\Re\left(p^{2}\right)>0,\Re(0+k+1)>0}} 
int(t*exp(- (p)^(2)* (t)^(2))*BesselI(0, a*t)*BesselK(0, a*t), t = 0..infinity) = (1)/(4*(p)^(2))*exp(((a)^(2))/(2*(p)^(2)))*BesselK(0, ((a)^(2))/(2*(p)^(2)))
 
Integrate[t*Exp[- (p)^(2)* (t)^(2)]*BesselI[0, a*t]*BesselK[0, a*t], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,4*(p)^(2)]*Exp[Divide[(a)^(2),2*(p)^(2)]]*BesselK[0, Divide[(a)^(2),2*(p)^(2)]]
 Failure Aborted Skipped - Because timed out Successful [Tested: 48]
10.44#Ex1 I ν ⁡ ( z ) = ∑ k = 0 ∞ z k k ! ⁢ J ν + k ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 superscript subscript 𝑘 0 superscript 𝑧 𝑘 𝑘 Bessel-J 𝜈 𝑘 𝑧 {\displaystyle{\displaystyle I_{\nu}\left(z\right)=\sum_{k=0}^{\infty}\frac{z^% {k}}{k!}J_{\nu+k}\left(z\right)}}
\modBesselI{\nu}@{z} = \sum_{k=0}^{\infty}\frac{z^{k}}{k!}\BesselJ{\nu+k}@{z}		 ℜ ⁡ ( ( ν + k ) + k + 1 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 𝑘 𝑘 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re((\nu+k)+k+1)>0,\Re(\nu+k+1)>0}} 
BesselI(nu, z) = sum(((z)^(k))/(factorial(k))*BesselJ(nu + k, z), k = 0..infinity)
 
BesselI[\[Nu], z] == Sum[Divide[(z)^(k),(k)!]*BesselJ[\[Nu]+ k, z], {k, 0, Infinity}, GenerateConditions->None]
 Failure Successful Skipped - Because timed out Successful [Tested: 70]
10.44#Ex2 J ν ⁡ ( z ) = ∑ k = 0 ∞ ( - 1 ) k ⁢ z k k ! ⁢ I ν + k ⁡ ( z ) Bessel-J 𝜈 𝑧 superscript subscript 𝑘 0 superscript 1 𝑘 superscript 𝑧 𝑘 𝑘 modified-Bessel-first-kind 𝜈 𝑘 𝑧 {\displaystyle{\displaystyle J_{\nu}\left(z\right)=\sum_{k=0}^{\infty}(-1)^{k}% \frac{z^{k}}{k!}I_{\nu+k}\left(z\right)}}
\BesselJ{\nu}@{z} = \sum_{k=0}^{\infty}(-1)^{k}\frac{z^{k}}{k!}\modBesselI{\nu+k}@{z}		 ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( ν + k ) + k + 1 ) > 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 𝑘 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((\nu+k)+k+1)>0}} 
BesselJ(nu, z) = sum((- 1)^(k)*((z)^(k))/(factorial(k))*BesselI(nu + k, z), k = 0..infinity)
 
BesselJ[\[Nu], z] == Sum[(- 1)^(k)*Divide[(z)^(k),(k)!]*BesselI[\[Nu]+ k, z], {k, 0, Infinity}, GenerateConditions->None]
 Failure Failure Skipped - Because timed out
Failed [70 / 70]
Result: Plus[Complex[0.4358908643715884, -0.07192294931339177], Times[-1.0, NSum[Times[Power[-1, k], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k], BesselI[Plus[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[Factorial[k], -1]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Plus[Complex[1.0679098760861825, 0.09257666026367889], Times[-1.0, NSum[Times[Power[-1, k], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k], BesselI[Plus[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[Factorial[k], -1]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.44.E4 ( 1 2 ⁢ z ) ν = ∑ k = 0 ∞ ( - 1 ) k ⁢ ( ν + 2 ⁢ k ) ⁢ Γ ⁡ ( ν + k ) k ! ⁢ I ν + 2 ⁢ k ⁡ ( z ) superscript 1 2 𝑧 𝜈 superscript subscript 𝑘 0 superscript 1 𝑘 𝜈 2 𝑘 Euler-Gamma 𝜈 𝑘 𝑘 modified-Bessel-first-kind 𝜈 2 𝑘 𝑧 {\displaystyle{\displaystyle\left(\tfrac{1}{2}z\right)^{\nu}=\sum_{k=0}^{% \infty}(-1)^{k}\frac{(\nu+2k)\Gamma\left(\nu+k\right)}{k!}I_{\nu+2k}\left(z% \right)}}
\left(\tfrac{1}{2}z\right)^{\nu} = \sum_{k=0}^{\infty}(-1)^{k}\frac{(\nu+2k)\EulerGamma@{\nu+k}}{k!}\modBesselI{\nu+2k}@{z}		 ℜ ⁡ ( ν + k ) > 0 , ℜ ⁡ ( ( ν + 2 ⁢ k ) + k + 1 ) > 0 formulae-sequence 𝜈 𝑘 0 𝜈 2 𝑘 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k)>0,\Re((\nu+2k)+k+1)>0}} 
((1)/(2)*z)^(nu) = sum((- 1)^(k)*((nu + 2*k)*GAMMA(nu + k))/(factorial(k))*BesselI(nu + 2*k, z), k = 0..infinity)
 
(Divide[1,2]*z)^\[Nu] == Sum[(- 1)^(k)*Divide[(\[Nu]+ 2*k)*Gamma[\[Nu]+ k],(k)!]*BesselI[\[Nu]+ 2*k, z], {k, 0, Infinity}, GenerateConditions->None]
 Failure Failure Manual Skip!
Failed [7 / 7]
Result: Plus[Complex[0.43301270189221935, 0.24999999999999997], Times[-1.0, NSum[Times[Power[-1, k], Plus[1, Times[2, k]], BesselI[Plus[1, Times[2, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[Factorial[k], -1], Gamma[Plus[1, k]]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 1]}

Result: Plus[Complex[-0.2499999999999999, 0.43301270189221935], Times[-1.0, NSum[Times[Power[-1, k], Plus[1, Times[2, k]], BesselI[Plus[1, Times[2, k]], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Power[Factorial[k], -1], Gamma[Plus[1, k]]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, 1]}

... skip entries to safe data
10.44.E5 K 0 ⁡ ( z ) = - ( ln ⁡ ( 1 2 ⁢ z ) + γ ) ⁢ I 0 ⁡ ( z ) + 2 ⁢ ∑ k = 1 ∞ I 2 ⁢ k ⁡ ( z ) k modified-Bessel-second-kind 0 𝑧 1 2 𝑧 modified-Bessel-first-kind 0 𝑧 2 superscript subscript 𝑘 1 modified-Bessel-first-kind 2 𝑘 𝑧 𝑘 {\displaystyle{\displaystyle K_{0}\left(z\right)=-\left(\ln\left(\tfrac{1}{2}z% \right)+\gamma\right)I_{0}\left(z\right)+2\sum_{k=1}^{\infty}\frac{I_{2k}\left% (z\right)}{k}}}
\modBesselK{0}@{z} = -\left(\ln@{\tfrac{1}{2}z}+\EulerConstant\right)\modBesselI{0}@{z}+2\sum_{k=1}^{\infty}\frac{\modBesselI{2k}@{z}}{k}		 ℜ ⁡ ( 0 + k + 1 ) > 0 , ℜ ⁡ ( ( 2 ⁢ k ) + k + 1 ) > 0 formulae-sequence 0 𝑘 1 0 2 𝑘 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0,\Re((2k)+k+1)>0}} 
BesselK(0, z) = -(ln((1)/(2)*z)+ gamma)*BesselI(0, z)+ 2*sum((BesselI(2*k, z))/(k), k = 1..infinity)
 
BesselK[0, z] == -(Log[Divide[1,2]*z]+ EulerGamma)*BesselI[0, z]+ 2*Sum[Divide[BesselI[2*k, z],k], {k, 1, Infinity}, GenerateConditions->None]
 Failure Successful Successful [Tested: 7] Successful [Tested: 7]
10.44.E6 K n ⁡ ( z ) = n ! ⁢ ( 1 2 ⁢ z ) - n 2 ⁢ ∑ k = 0 n - 1 ( - 1 ) k ⁢ ( 1 2 ⁢ z ) k ⁢ I k ⁡ ( z ) k ! ⁢ ( n - k ) + ( - 1 ) n - 1 ⁢ ( ln ⁡ ( 1 2 ⁢ z ) - ψ ⁡ ( n + 1 ) ) ⁢ I n ⁡ ( z ) + ( - 1 ) n ⁢ ∑ k = 1 ∞ ( n + 2 ⁢ k ) ⁢ I n + 2 ⁢ k ⁡ ( z ) k ⁢ ( n + k ) modified-Bessel-second-kind 𝑛 𝑧 𝑛 superscript 1 2 𝑧 𝑛 2 superscript subscript 𝑘 0 𝑛 1 superscript 1 𝑘 superscript 1 2 𝑧 𝑘 modified-Bessel-first-kind 𝑘 𝑧 𝑘 𝑛 𝑘 superscript 1 𝑛 1 1 2 𝑧 digamma 𝑛 1 modified-Bessel-first-kind 𝑛 𝑧 superscript 1 𝑛 superscript subscript 𝑘 1 𝑛 2 𝑘 modified-Bessel-first-kind 𝑛 2 𝑘 𝑧 𝑘 𝑛 𝑘 {\displaystyle{\displaystyle K_{n}\left(z\right)=\frac{n!(\tfrac{1}{2}z)^{-n}}% {2}\sum_{k=0}^{n-1}(-1)^{k}\frac{(\tfrac{1}{2}z)^{k}I_{k}\left(z\right)}{k!(n-% k)}+(-1)^{n-1}\left(\ln\left(\tfrac{1}{2}z\right)-\psi\left(n+1\right)\right)I% _{n}\left(z\right)+(-1)^{n}\sum_{k=1}^{\infty}\frac{(n+2k)I_{n+2k}\left(z% \right)}{k(n+k)}}}
\modBesselK{n}@{z} = \frac{n!(\tfrac{1}{2}z)^{-n}}{2}\sum_{k=0}^{n-1}(-1)^{k}\frac{(\tfrac{1}{2}z)^{k}\modBesselI{k}@{z}}{k!(n-k)}+(-1)^{n-1}\left(\ln@{\tfrac{1}{2}z}-\digamma@{n+1}\right)\modBesselI{n}@{z}+(-1)^{n}\sum_{k=1}^{\infty}\frac{(n+2k)\modBesselI{n+2k}@{z}}{k(n+k)}		 ℜ ⁡ ( n + k + 1 ) > 0 , ℜ ⁡ ( k + k + 1 ) > 0 , ℜ ⁡ ( ( n + 2 ⁢ k ) + k + 1 ) > 0 formulae-sequence 𝑛 𝑘 1 0 formulae-sequence 𝑘 𝑘 1 0 𝑛 2 𝑘 𝑘 1 0 {\displaystyle{\displaystyle\Re(n+k+1)>0,\Re(k+k+1)>0,\Re((n+2k)+k+1)>0}} 
BesselK(n, z) = (factorial(n)*((1)/(2)*z)^(- n))/(2)*sum((- 1)^(k)*(((1)/(2)*z)^(k)* BesselI(k, z))/(factorial(k)*(n - k)), k = 0..n - 1)+(- 1)^(n - 1)*(ln((1)/(2)*z)- Psi(n + 1))*BesselI(n, z)+(- 1)^(n)* sum(((n + 2*k)*BesselI(n + 2*k, z))/(k*(n + k)), k = 1..infinity)
 
BesselK[n, z] == Divide[(n)!*(Divide[1,2]*z)^(- n),2]*Sum[(- 1)^(k)*Divide[(Divide[1,2]*z)^(k)* BesselI[k, z],(k)!*(n - k)], {k, 0, n - 1}, GenerateConditions->None]+(- 1)^(n - 1)*(Log[Divide[1,2]*z]- PolyGamma[n + 1])*BesselI[n, z]+(- 1)^(n)* Sum[Divide[(n + 2*k)*BesselI[n + 2*k, z],k*(n + k)], {k, 1, Infinity}, GenerateConditions->None]
 Failure Aborted Manual Skip!
Failed [21 / 21]
Result: Plus[Complex[1.084080291505059, -0.3914662527648858], NSum[Times[Power[k, -1], Power[Plus[1, k], -1], Plus[1, Times[2, k]], BesselI[Plus[1, Times[2, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]], Times[Complex[-0.8660254037844387, 0.49999999999999994], DifferenceRoot[Function[{, }, {Equal[Plus[Times[-1, Plus[Times[-1, ], 1], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], []], Times[Plus[4, Times[12, ], Times[12, Power[, 2]], Times[4, Power[, 3]], Times[-4, 1], Times[-8, , 1], Times[-4, Power[, 2], 1], Times[-1, , Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[1, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], [Plus[1, ]]], Times[4, Plus[1, ], Plus[-5, Times[-6, ], Times[-2, Power[, 2]], Times[3, 1], Times[2, , 1]], [Plus[2, ]]], Times[-4, Plus[1, ], Plus[2, ], Plus[-2, Times[-1, ], 1], [Plus[3, ]<syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.001928095904955185, 0.0030033056761246957], Times[-1.0, NSum[Times[Power[k, -1], Power[Plus[2, k], -1], Plus[2, Times[2, k]], BesselI[Plus[2, Times[2, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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