Bessel Functions - 10.39 Relations to Other Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
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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
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