Bessel Functions - 10.32 Integral Representations

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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
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