Confluent Hypergeometric Functions - 13.16 Integral Representations

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13.16.E1 M κ , μ ⁡ ( z ) = Γ ⁡ ( 1 + 2 ⁢ μ ) ⁢ z μ + 1 2 ⁢ 2 - 2 ⁢ μ Γ ⁡ ( 1 2 + μ - κ ) ⁢ Γ ⁡ ( 1 2 + μ + κ ) ⁢ ∫ - 1 1 e 1 2 ⁢ z ⁢ t ⁢ ( 1 + t ) μ - 1 2 - κ ⁢ ( 1 - t ) μ - 1 2 + κ ⁢ d t Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 Euler-Gamma 1 2 𝜇 superscript 𝑧 𝜇 1 2 superscript 2 2 𝜇 Euler-Gamma 1 2 𝜇 𝜅 Euler-Gamma 1 2 𝜇 𝜅 superscript subscript 1 1 superscript 𝑒 1 2 𝑧 𝑡 superscript 1 𝑡 𝜇 1 2 𝜅 superscript 1 𝑡 𝜇 1 2 𝜅 𝑡 {\displaystyle{\displaystyle M_{\kappa,\mu}\left(z\right)=\frac{\Gamma\left(1+% 2\mu\right)z^{\mu+\frac{1}{2}}2^{-2\mu}}{\Gamma\left(\frac{1}{2}+\mu-\kappa% \right)\Gamma\left(\frac{1}{2}+\mu+\kappa\right)}\*\int_{-1}^{1}e^{\frac{1}{2}% zt}(1+t)^{\mu-\frac{1}{2}-\kappa}(1-t)^{\mu-\frac{1}{2}+\kappa}\mathrm{d}t}}
\WhittakerconfhyperM{\kappa}{\mu}@{z} = \frac{\EulerGamma@{1+2\mu}z^{\mu+\frac{1}{2}}2^{-2\mu}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}+\mu+\kappa}}\*\int_{-1}^{1}e^{\frac{1}{2}zt}(1+t)^{\mu-\frac{1}{2}-\kappa}(1-t)^{\mu-\frac{1}{2}+\kappa}\diff{t}		 ℜ ⁡ μ + 1 2 > | ℜ ⁡ κ | , ℜ ⁡ ( 1 + 2 ⁢ μ ) > 0 , ℜ ⁡ ( 1 2 + μ - κ ) > 0 , ℜ ⁡ ( 1 2 + μ + κ ) > 0 formulae-sequence 𝜇 1 2 𝜅 formulae-sequence 1 2 𝜇 0 formulae-sequence 1 2 𝜇 𝜅 0 1 2 𝜇 𝜅 0 {\displaystyle{\displaystyle\Re\mu+\tfrac{1}{2}>\left|\Re\kappa\right|,\Re(1+2% \mu)>0,\Re(\frac{1}{2}+\mu-\kappa)>0,\Re(\frac{1}{2}+\mu+\kappa)>0}} 
WhittakerM(kappa, mu, z) = (GAMMA(1 + 2*mu)*(z)^(mu +(1)/(2))* (2)^(- 2*mu))/(GAMMA((1)/(2)+ mu - kappa)*GAMMA((1)/(2)+ mu + kappa))* int(exp((1)/(2)*z*t)*(1 + t)^(mu -(1)/(2)- kappa)*(1 - t)^(mu -(1)/(2)+ kappa), t = - 1..1)

WhittakerM[\[Kappa], \[Mu], z] == Divide[Gamma[1 + 2*\[Mu]]*(z)^(\[Mu]+Divide[1,2])* (2)^(- 2*\[Mu]),Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]*Gamma[Divide[1,2]+ \[Mu]+ \[Kappa]]]* Integrate[Exp[Divide[1,2]*z*t]*(1 + t)^(\[Mu]-Divide[1,2]- \[Kappa])*(1 - t)^(\[Mu]-Divide[1,2]+ \[Kappa]), {t, - 1, 1}, GenerateConditions->None]
Failure Successful Skipped - Because timed out Successful [Tested: 252]
13.16.E2 M κ , μ ⁡ ( z ) = Γ ⁡ ( 1 + 2 ⁢ μ ) ⁢ z λ Γ ⁡ ( 1 + 2 ⁢ μ - 2 ⁢ λ ) ⁢ Γ ⁡ ( 2 ⁢ λ ) ⁢ ∫ 0 1 M κ - λ , μ - λ ⁡ ( z ⁢ t ) ⁢ e 1 2 ⁢ z ⁢ ( t - 1 ) ⁢ t μ - λ - 1 2 ⁢ ( 1 - t ) 2 ⁢ λ - 1 ⁢ d t Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 Euler-Gamma 1 2 𝜇 superscript 𝑧 𝜆 Euler-Gamma 1 2 𝜇 2 𝜆 Euler-Gamma 2 𝜆 superscript subscript 0 1 Whittaker-confluent-hypergeometric-M 𝜅 𝜆 𝜇 𝜆 𝑧 𝑡 superscript 𝑒 1 2 𝑧 𝑡 1 superscript 𝑡 𝜇 𝜆 1 2 superscript 1 𝑡 2 𝜆 1 𝑡 {\displaystyle{\displaystyle M_{\kappa,\mu}\left(z\right)=\frac{\Gamma\left(1+% 2\mu\right)z^{\lambda}}{\Gamma\left(1+2\mu-2\lambda\right)\Gamma\left(2\lambda% \right)}\*\int_{0}^{1}M_{\kappa-\lambda,\mu-\lambda}\left(zt\right)e^{\frac{1}% {2}z(t-1)}t^{\mu-\lambda-\frac{1}{2}}{(1-t)^{2\lambda-1}}\mathrm{d}t}}
\WhittakerconfhyperM{\kappa}{\mu}@{z} = \frac{\EulerGamma@{1+2\mu}z^{\lambda}}{\EulerGamma@{1+2\mu-2\lambda}\EulerGamma@{2\lambda}}\*\int_{0}^{1}\WhittakerconfhyperM{\kappa-\lambda}{\mu-\lambda}@{zt}e^{\frac{1}{2}z(t-1)}t^{\mu-\lambda-\frac{1}{2}}{(1-t)^{2\lambda-1}}\diff{t}		 ℜ ⁡ μ + 1 2 > ℜ ⁡ λ , ℜ ⁡ λ > 0 , ℜ ⁡ ( 1 + 2 ⁢ μ ) > 0 , ℜ ⁡ ( 1 + 2 ⁢ μ - 2 ⁢ λ ) > 0 , ℜ ⁡ ( 2 ⁢ λ ) > 0 formulae-sequence 𝜇 1 2 𝜆 formulae-sequence 𝜆 0 formulae-sequence 1 2 𝜇 0 formulae-sequence 1 2 𝜇 2 𝜆 0 2 𝜆 0 {\displaystyle{\displaystyle\Re\mu+\tfrac{1}{2}>\Re\lambda,\Re\lambda>0,\Re(1+% 2\mu)>0,\Re(1+2\mu-2\lambda)>0,\Re(2\lambda)>0}} 
WhittakerM(kappa, mu, z) = (GAMMA(1 + 2*mu)*(z)^(lambda))/(GAMMA(1 + 2*mu - 2*lambda)*GAMMA(2*lambda))* int(WhittakerM(kappa - lambda, mu - lambda, z*t)*exp((1)/(2)*z*(t - 1))*(t)^(mu - lambda -(1)/(2))*(1 - t)^(2*lambda - 1), t = 0..1)

WhittakerM[\[Kappa], \[Mu], z] == Divide[Gamma[1 + 2*\[Mu]]*(z)^\[Lambda],Gamma[1 + 2*\[Mu]- 2*\[Lambda]]*Gamma[2*\[Lambda]]]* Integrate[WhittakerM[\[Kappa]- \[Lambda], \[Mu]- \[Lambda], z*t]*Exp[Divide[1,2]*z*(t - 1)]*(t)^(\[Mu]- \[Lambda]-Divide[1,2])*(1 - t)^(2*\[Lambda]- 1), {t, 0, 1}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
13.16.E3 1 Γ ⁡ ( 1 + 2 ⁢ μ ) ⁢ M κ , μ ⁡ ( z ) = z ⁢ e 1 2 ⁢ z Γ ⁡ ( 1 2 + μ + κ ) ⁢ ∫ 0 ∞ e - t ⁢ t κ - 1 2 ⁢ J 2 ⁢ μ ⁡ ( 2 ⁢ z ⁢ t ) ⁢ d t 1 Euler-Gamma 1 2 𝜇 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 𝑧 superscript 𝑒 1 2 𝑧 Euler-Gamma 1 2 𝜇 𝜅 superscript subscript 0 superscript 𝑒 𝑡 superscript 𝑡 𝜅 1 2 Bessel-J 2 𝜇 2 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle\frac{1}{\Gamma\left(1+2\mu\right)}M_{\kappa,\mu}% \left(z\right)=\frac{\sqrt{z}e^{\frac{1}{2}z}}{\Gamma\left(\frac{1}{2}+\mu+% \kappa\right)}\int_{0}^{\infty}e^{-t}t^{\kappa-\frac{1}{2}}J_{2\mu}\left(2% \sqrt{zt}\right)\mathrm{d}t}}
\frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{z} = \frac{\sqrt{z}e^{\frac{1}{2}z}}{\EulerGamma@{\frac{1}{2}+\mu+\kappa}}\int_{0}^{\infty}e^{-t}t^{\kappa-\frac{1}{2}}\BesselJ{2\mu}@{2\sqrt{zt}}\diff{t}		 ℜ ⁡ ( κ + μ ) + 1 2 > 0 , ℜ ⁡ ( ( 2 ⁢ μ ) + k + 1 ) > 0 , ℜ ⁡ ( 1 + 2 ⁢ μ ) > 0 , ℜ ⁡ ( 1 2 + μ + κ ) > 0 formulae-sequence 𝜅 𝜇 1 2 0 formulae-sequence 2 𝜇 𝑘 1 0 formulae-sequence 1 2 𝜇 0 1 2 𝜇 𝜅 0 {\displaystyle{\displaystyle\Re\left(\kappa+\mu\right)+\tfrac{1}{2}>0,\Re((2% \mu)+k+1)>0,\Re(1+2\mu)>0,\Re(\frac{1}{2}+\mu+\kappa)>0}} 
(1)/(GAMMA(1 + 2*mu))*WhittakerM(kappa, mu, z) = (sqrt(z)*exp((1)/(2)*z))/(GAMMA((1)/(2)+ mu + kappa))*int(exp(- t)*(t)^(kappa -(1)/(2))* BesselJ(2*mu, 2*sqrt(z*t)), t = 0..infinity)

Divide[1,Gamma[1 + 2*\[Mu]]]*WhittakerM[\[Kappa], \[Mu], z] == Divide[Sqrt[z]*Exp[Divide[1,2]*z],Gamma[Divide[1,2]+ \[Mu]+ \[Kappa]]]*Integrate[Exp[- t]*(t)^(\[Kappa]-Divide[1,2])* BesselJ[2*\[Mu], 2*Sqrt[z*t]], {t, 0, Infinity}, GenerateConditions->None]
Successful Aborted - Skipped - Because timed out
13.16.E4 1 Γ ⁡ ( 1 + 2 ⁢ μ ) ⁢ M κ , μ ⁡ ( z ) = z ⁢ e - 1 2 ⁢ z Γ ⁡ ( 1 2 + μ - κ ) ⁢ ∫ 0 ∞ e - t ⁢ t - κ - 1 2 ⁢ I 2 ⁢ μ ⁡ ( 2 ⁢ z ⁢ t ) ⁢ d t 1 Euler-Gamma 1 2 𝜇 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 𝑧 superscript 𝑒 1 2 𝑧 Euler-Gamma 1 2 𝜇 𝜅 superscript subscript 0 superscript 𝑒 𝑡 superscript 𝑡 𝜅 1 2 modified-Bessel-first-kind 2 𝜇 2 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle\frac{1}{\Gamma\left(1+2\mu\right)}M_{\kappa,\mu}% \left(z\right)=\frac{\sqrt{z}e^{-\frac{1}{2}z}}{\Gamma\left(\frac{1}{2}+\mu-% \kappa\right)}\*\int_{0}^{\infty}e^{-t}t^{-\kappa-\frac{1}{2}}I_{2\mu}\left(2% \sqrt{zt}\right)\mathrm{d}t}}
\frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{z} = \frac{\sqrt{z}e^{-\frac{1}{2}z}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*\int_{0}^{\infty}e^{-t}t^{-\kappa-\frac{1}{2}}\modBesselI{2\mu}@{2\sqrt{zt}}\diff{t}		 ℜ ⁡ ( κ - μ ) - 1 2 < 0 , ℜ ⁡ ( 1 + 2 ⁢ μ ) > 0 , ℜ ⁡ ( 1 2 + μ - κ ) > 0 , ℜ ⁡ ( ( 2 ⁢ μ ) + k + 1 ) > 0 formulae-sequence 𝜅 𝜇 1 2 0 formulae-sequence 1 2 𝜇 0 formulae-sequence 1 2 𝜇 𝜅 0 2 𝜇 𝑘 1 0 {\displaystyle{\displaystyle\Re(\kappa-\mu)-\tfrac{1}{2}<0,\Re(1+2\mu)>0,\Re(% \frac{1}{2}+\mu-\kappa)>0,\Re((2\mu)+k+1)>0}} 
(1)/(GAMMA(1 + 2*mu))*WhittakerM(kappa, mu, z) = (sqrt(z)*exp(-(1)/(2)*z))/(GAMMA((1)/(2)+ mu - kappa))* int(exp(- t)*(t)^(- kappa -(1)/(2))* BesselI(2*mu, 2*sqrt(z*t)), t = 0..infinity)

Divide[1,Gamma[1 + 2*\[Mu]]]*WhittakerM[\[Kappa], \[Mu], z] == Divide[Sqrt[z]*Exp[-Divide[1,2]*z],Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]]* Integrate[Exp[- t]*(t)^(- \[Kappa]-Divide[1,2])* BesselI[2*\[Mu], 2*Sqrt[z*t]], {t, 0, Infinity}, GenerateConditions->None]
Failure Successful
Failed [42 / 300]
Result: .5483729950e-2+.5411197480e-1*I
Test Values: {kappa = -3/2, mu = 2, z = 1/2*3^(1/2)+1/2*I}


Result: .2482822497e-1-.2550894001e-1*I
Test Values: {kappa = -3/2, mu = 2, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Successful [Tested: 300]
13.16.E5 W κ , μ ⁡ ( z ) = z μ + 1 2 ⁢ 2 - 2 ⁢ μ Γ ⁡ ( 1 2 + μ - κ ) ⁢ ∫ 1 ∞ e - 1 2 ⁢ z ⁢ t ⁢ ( t - 1 ) μ - 1 2 - κ ⁢ ( t + 1 ) μ - 1 2 + κ ⁢ d t Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 superscript 𝑧 𝜇 1 2 superscript 2 2 𝜇 Euler-Gamma 1 2 𝜇 𝜅 superscript subscript 1 superscript 𝑒 1 2 𝑧 𝑡 superscript 𝑡 1 𝜇 1 2 𝜅 superscript 𝑡 1 𝜇 1 2 𝜅 𝑡 {\displaystyle{\displaystyle W_{\kappa,\mu}\left(z\right)=\frac{z^{\mu+\frac{1% }{2}}2^{-2\mu}}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}\*\int_{1}^{\infty}e% ^{-\frac{1}{2}zt}(t-1)^{\mu-\frac{1}{2}-\kappa}(t+1)^{\mu-\frac{1}{2}+\kappa}% \mathrm{d}t}}
\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{z^{\mu+\frac{1}{2}}2^{-2\mu}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*\int_{1}^{\infty}e^{-\frac{1}{2}zt}(t-1)^{\mu-\frac{1}{2}-\kappa}(t+1)^{\mu-\frac{1}{2}+\kappa}\diff{t}		 ℜ ⁡ μ + 1 2 > ℜ ⁡ κ , | ph ⁡ z | < 1 2 ⁢ π , ℜ ⁡ ( 1 2 + μ - κ ) > 0 formulae-sequence 𝜇 1 2 𝜅 formulae-sequence phase 𝑧 1 2 𝜋 1 2 𝜇 𝜅 0 {\displaystyle{\displaystyle\Re\mu+\tfrac{1}{2}>\Re\kappa,|\operatorname{ph}{z% }|<\frac{1}{2}\pi,\Re(\frac{1}{2}+\mu-\kappa)>0}} 
WhittakerW(kappa, mu, z) = ((z)^(mu +(1)/(2))* (2)^(- 2*mu))/(GAMMA((1)/(2)+ mu - kappa))* int(exp(-(1)/(2)*z*t)*(t - 1)^(mu -(1)/(2)- kappa)*(t + 1)^(mu -(1)/(2)+ kappa), t = 1..infinity)

WhittakerW[\[Kappa], \[Mu], z] == Divide[(z)^(\[Mu]+Divide[1,2])* (2)^(- 2*\[Mu]),Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]]* Integrate[Exp[-Divide[1,2]*z*t]*(t - 1)^(\[Mu]-Divide[1,2]- \[Kappa])*(t + 1)^(\[Mu]-Divide[1,2]+ \[Kappa]), {t, 1, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Successful [Tested: 300]
13.16.E6 W κ , μ ⁡ ( z ) = e - 1 2 ⁢ z ⁢ z κ + 1 Γ ⁡ ( 1 2 + μ - κ ) ⁢ Γ ⁡ ( 1 2 - μ - κ ) ⁢ ∫ 0 ∞ W - κ , μ ⁡ ( t ) ⁢ e - 1 2 ⁢ t ⁢ t - κ - 1 t + z ⁢ d t Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜅 1 Euler-Gamma 1 2 𝜇 𝜅 Euler-Gamma 1 2 𝜇 𝜅 superscript subscript 0 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑡 superscript 𝑒 1 2 𝑡 superscript 𝑡 𝜅 1 𝑡 𝑧 𝑡 {\displaystyle{\displaystyle W_{\kappa,\mu}\left(z\right)=\frac{e^{-\frac{1}{2% }z}z^{\kappa+1}}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}% {2}-\mu-\kappa\right)}\*\int_{0}^{\infty}\frac{W_{-\kappa,\mu}\left(t\right)e^% {-\frac{1}{2}t}t^{-\kappa-1}}{t+z}\mathrm{d}t}}
\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{e^{-\frac{1}{2}z}z^{\kappa+1}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}\*\int_{0}^{\infty}\frac{\WhittakerconfhyperW{-\kappa}{\mu}@{t}e^{-\frac{1}{2}t}t^{-\kappa-1}}{t+z}\diff{t}		 | ph z | < π , ℜ ⁡ ( 1 2 + μ - κ ) > max ( 2 ℜ ⁡ μ , ℜ ⁡ ( 1 2 + μ - κ ) > 0 , ℜ ⁡ ( 1 2 - μ - κ ) > 0 fragments | phase z | π , 1 2 𝜇 𝜅 fragments ( 2 𝜇 , 1 2 𝜇 𝜅 0 , 1 2 𝜇 𝜅 0 {\displaystyle{\displaystyle|\operatorname{ph}{z}|<\pi,\Re\left(\frac{1}{2}+% \mu-\kappa\right)>\max\left(2\Re\mu,\Re(\frac{1}{2}+\mu-\kappa)>0,\Re(\frac{1}% {2}-\mu-\kappa)>0}\)\@add@PDF@RDFa@triples\end{document}} 
WhittakerW(kappa, mu, z) = (exp(-(1)/(2)*z)*(z)^(kappa + 1))/(GAMMA((1)/(2)+ mu - kappa)*GAMMA((1)/(2)- mu - kappa))* int((WhittakerW(- kappa, mu, t)*exp(-(1)/(2)*t)*(t)^(- kappa - 1))/(t + z), t = 0..infinity)

WhittakerW[\[Kappa], \[Mu], z] == Divide[Exp[-Divide[1,2]*z]*(z)^(\[Kappa]+ 1),Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]*Gamma[Divide[1,2]- \[Mu]- \[Kappa]]]* Integrate[Divide[WhittakerW[- \[Kappa], \[Mu], t]*Exp[-Divide[1,2]*t]*(t)^(- \[Kappa]- 1),t + z], {t, 0, Infinity}, GenerateConditions->None]
Failure Aborted Manual Skip! Successful [Tested: 300]
13.16.E7 W κ , μ ⁡ ( z ) = ( - 1 ) n ⁢ e - 1 2 ⁢ z ⁢ z 1 2 - μ - n Γ ⁡ ( 1 + 2 ⁢ μ ) ⁢ Γ ⁡ ( 1 2 - μ - κ ) ⁢ ∫ 0 ∞ M - κ , μ ⁡ ( t ) ⁢ e - 1 2 ⁢ t ⁢ t n + μ - 1 2 t + z ⁢ d t Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 superscript 1 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 1 2 𝜇 𝑛 Euler-Gamma 1 2 𝜇 Euler-Gamma 1 2 𝜇 𝜅 superscript subscript 0 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑡 superscript 𝑒 1 2 𝑡 superscript 𝑡 𝑛 𝜇 1 2 𝑡 𝑧 𝑡 {\displaystyle{\displaystyle W_{\kappa,\mu}\left(z\right)=\frac{(-1)^{n}e^{-% \frac{1}{2}z}z^{\frac{1}{2}-\mu-n}}{\Gamma\left(1+2\mu\right)\Gamma\left(\frac% {1}{2}-\mu-\kappa\right)}\*\int_{0}^{\infty}\frac{M_{-\kappa,\mu}\left(t\right% )e^{-\frac{1}{2}t}t^{n+\mu-\frac{1}{2}}}{t+z}\mathrm{d}t}}
\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{(-1)^{n}e^{-\frac{1}{2}z}z^{\frac{1}{2}-\mu-n}}{\EulerGamma@{1+2\mu}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}\*\int_{0}^{\infty}\frac{\WhittakerconfhyperM{-\kappa}{\mu}@{t}e^{-\frac{1}{2}t}t^{n+\mu-\frac{1}{2}}}{t+z}\diff{t}		 | ph ⁡ z | < π , - ℜ ⁡ ( 1 + 2 ⁢ μ ) < n , n < | ℜ ⁡ μ | + ℜ ⁡ κ , | ℜ ⁡ μ | + ℜ ⁡ κ < 1 2 , ℜ ⁡ ( 1 + 2 ⁢ μ ) > 0 , ℜ ⁡ ( 1 2 - μ - κ ) > 0 formulae-sequence phase 𝑧 𝜋 formulae-sequence 1 2 𝜇 𝑛 formulae-sequence 𝑛 𝜇 𝜅 formulae-sequence 𝜇 𝜅 1 2 formulae-sequence 1 2 𝜇 0 1 2 𝜇 𝜅 0 {\displaystyle{\displaystyle|\operatorname{ph}z|<\pi,-\Re\left(1+2\mu\right)<n% ,n<\left|\Re\mu\right|+\Re\kappa,\left|\Re\mu\right|+\Re\kappa<\tfrac{1}{2},% \Re(1+2\mu)>0,\Re(\frac{1}{2}-\mu-\kappa)>0}} 
WhittakerW(kappa, mu, z) = ((- 1)^(n)* exp(-(1)/(2)*z)*(z)^((1)/(2)- mu - n))/(GAMMA(1 + 2*mu)*GAMMA((1)/(2)- mu - kappa))* int((WhittakerM(- kappa, mu, t)*exp(-(1)/(2)*t)*(t)^(n + mu -(1)/(2)))/(t + z), t = 0..infinity)

WhittakerW[\[Kappa], \[Mu], z] == Divide[(- 1)^(n)* Exp[-Divide[1,2]*z]*(z)^(Divide[1,2]- \[Mu]- n),Gamma[1 + 2*\[Mu]]*Gamma[Divide[1,2]- \[Mu]- \[Kappa]]]* Integrate[Divide[WhittakerM[- \[Kappa], \[Mu], t]*Exp[-Divide[1,2]*t]*(t)^(n + \[Mu]-Divide[1,2]),t + z], {t, 0, Infinity}, GenerateConditions->None]
Failure Aborted Manual Skip! Skipped - Because timed out
13.16.E8 W κ , μ ⁡ ( z ) = 2 ⁢ z ⁢ e - 1 2 ⁢ z Γ ⁡ ( 1 2 + μ - κ ) ⁢ Γ ⁡ ( 1 2 - μ - κ ) ⁢ ∫ 0 ∞ e - t ⁢ t - κ - 1 2 ⁢ K 2 ⁢ μ ⁡ ( 2 ⁢ z ⁢ t ) ⁢ d t Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 2 𝑧 superscript 𝑒 1 2 𝑧 Euler-Gamma 1 2 𝜇 𝜅 Euler-Gamma 1 2 𝜇 𝜅 superscript subscript 0 superscript 𝑒 𝑡 superscript 𝑡 𝜅 1 2 modified-Bessel-second-kind 2 𝜇 2 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle W_{\kappa,\mu}\left(z\right)=\frac{2\sqrt{z}e^{-% \frac{1}{2}z}}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}{2% }-\mu-\kappa\right)}\*\int_{0}^{\infty}e^{-t}t^{-\kappa-\frac{1}{2}}K_{2\mu}% \left(2\sqrt{zt}\right)\mathrm{d}t}}
\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{2\sqrt{z}e^{-\frac{1}{2}z}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}\*\int_{0}^{\infty}e^{-t}t^{-\kappa-\frac{1}{2}}\modBesselK{2\mu}@{2\sqrt{zt}}\diff{t}		 ℜ ⁡ ( μ - κ ) + 1 2 > 0 , ℜ ⁡ ( 1 2 + μ - κ ) > 0 , ℜ ⁡ ( 1 2 - μ - κ ) > 0 formulae-sequence 𝜇 𝜅 1 2 0 formulae-sequence 1 2 𝜇 𝜅 0 1 2 𝜇 𝜅 0 {\displaystyle{\displaystyle\Re\left(\mu-\kappa\right)+\tfrac{1}{2}>0,\Re(% \frac{1}{2}+\mu-\kappa)>0,\Re(\frac{1}{2}-\mu-\kappa)>0}} 
WhittakerW(kappa, mu, z) = (2*sqrt(z)*exp(-(1)/(2)*z))/(GAMMA((1)/(2)+ mu - kappa)*GAMMA((1)/(2)- mu - kappa))* int(exp(- t)*(t)^(- kappa -(1)/(2))* BesselK(2*mu, 2*sqrt(z*t)), t = 0..infinity)

WhittakerW[\[Kappa], \[Mu], z] == Divide[2*Sqrt[z]*Exp[-Divide[1,2]*z],Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]*Gamma[Divide[1,2]- \[Mu]- \[Kappa]]]* Integrate[Exp[- t]*(t)^(- \[Kappa]-Divide[1,2])* BesselK[2*\[Mu], 2*Sqrt[z*t]], {t, 0, Infinity}, GenerateConditions->None]
Successful Aborted - Successful [Tested: 252]
13.16.E9 W κ , μ ⁡ ( z ) = e - 1 2 ⁢ z ⁢ z κ + c ⁢ ∫ 0 ∞ e - z ⁢ t ⁢ t c - 1 ⁢ 𝐅 1 2 ⁡ ( 1 2 + μ - κ , 1 2 - μ - κ c ; - t ) ⁢ d t Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜅 𝑐 superscript subscript 0 superscript 𝑒 𝑧 𝑡 superscript 𝑡 𝑐 1 hypergeometric-bold-pFq 2 1 1 2 𝜇 𝜅 1 2 𝜇 𝜅 𝑐 𝑡 𝑡 {\displaystyle{\displaystyle W_{\kappa,\mu}\left(z\right)=e^{-\frac{1}{2}z}z^{% \kappa+c}\*\int_{0}^{\infty}e^{-zt}t^{c-1}{{}_{2}{\mathbf{F}}_{1}}\left({% \tfrac{1}{2}+\mu-\kappa,\tfrac{1}{2}-\mu-\kappa\atop c};-t\right)\mathrm{d}t}}
\WhittakerconfhyperW{\kappa}{\mu}@{z} = e^{-\frac{1}{2}z}z^{\kappa+c}\*\int_{0}^{\infty}e^{-zt}t^{c-1}\genhyperOlverF{2}{1}@@{\tfrac{1}{2}+\mu-\kappa,\tfrac{1}{2}-\mu-\kappa}{c}{-t}\diff{t}		 | ph ⁡ z | < 1 2 ⁢ π phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}{z}|<\frac{1}{2}\pi}} 
WhittakerW(kappa, mu, z) = exp(-(1)/(2)*z)*(z)^(kappa + c)* int(exp(- z*t)*(t)^(c - 1)* hypergeom([(1)/(2)+ mu - kappa ,(1)/(2)- mu - kappa], [c], - t), t = 0..infinity)

WhittakerW[\[Kappa], \[Mu], z] == Exp[-Divide[1,2]*z]*(z)^(\[Kappa]+ c)* Integrate[Exp[- z*t]*(t)^(c - 1)* HypergeometricPFQRegularized[{Divide[1,2]+ \[Mu]- \[Kappa],Divide[1,2]- \[Mu]- \[Kappa]}, {c}, - t], {t, 0, Infinity}, GenerateConditions->None]
Failure Aborted Manual Skip! Skipped - Because timed out
13.16.E10 1 Γ ⁡ ( 1 + 2 ⁢ μ ) ⁢ M κ , μ ⁡ ( e + π ⁢ i ⁢ z ) = e 1 2 ⁢ z + ( 1 2 + μ ) ⁢ π ⁢ i 2 ⁢ π ⁢ i ⁢ Γ ⁡ ( 1 2 + μ - κ ) ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ Γ ⁡ ( t - κ ) ⁢ Γ ⁡ ( 1 2 + μ - t ) Γ ⁡ ( 1 2 + μ + t ) ⁢ z t ⁢ d t 1 Euler-Gamma 1 2 𝜇 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 superscript 𝑒 𝜋 imaginary-unit 𝑧 superscript 𝑒 1 2 𝑧 1 2 𝜇 𝜋 imaginary-unit 2 𝜋 imaginary-unit Euler-Gamma 1 2 𝜇 𝜅 superscript subscript imaginary-unit imaginary-unit Euler-Gamma 𝑡 𝜅 Euler-Gamma 1 2 𝜇 𝑡 Euler-Gamma 1 2 𝜇 𝑡 superscript 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle\frac{1}{\Gamma\left(1+2\mu\right)}M_{\kappa,\mu}% \left(e^{+\pi\mathrm{i}}z\right)=\frac{e^{\frac{1}{2}z+(\frac{1}{2}+\mu)\pi% \mathrm{i}}}{2\pi\mathrm{i}\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}\*\int_{-% \mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(t-\kappa\right)\Gamma% \left(\frac{1}{2}+\mu-t\right)}{\Gamma\left(\frac{1}{2}+\mu+t\right)}z^{t}% \mathrm{d}t}}
\frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{e^{+\pi\iunit}z} = \frac{e^{\frac{1}{2}z+(\frac{1}{2}+\mu)\pi\iunit}}{2\pi\iunit\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{t-\kappa}\EulerGamma@{\frac{1}{2}+\mu-t}}{\EulerGamma@{\frac{1}{2}+\mu+t}}z^{t}\diff{t}		 | ph ⁡ z | < 1 2 ⁢ π , ℜ ⁡ ( 1 + 2 ⁢ μ ) > 0 , ℜ ⁡ ( 1 2 + μ - κ ) > 0 , ℜ ⁡ ( t - κ ) > 0 , ℜ ⁡ ( 1 2 + μ - t ) > 0 , ℜ ⁡ ( 1 2 + μ + t ) > 0 formulae-sequence phase 𝑧 1 2 𝜋 formulae-sequence 1 2 𝜇 0 formulae-sequence 1 2 𝜇 𝜅 0 formulae-sequence 𝑡 𝜅 0 formulae-sequence 1 2 𝜇 𝑡 0 1 2 𝜇 𝑡 0 {\displaystyle{\displaystyle|\operatorname{ph}{z}|<\tfrac{1}{2}\pi,\Re(1+2\mu)% >0,\Re(\frac{1}{2}+\mu-\kappa)>0,\Re(t-\kappa)>0,\Re(\frac{1}{2}+\mu-t)>0,\Re(% \frac{1}{2}+\mu+t)>0}} 
(1)/(GAMMA(1 + 2*mu))*WhittakerM(kappa, mu, exp(+ Pi*I)*z) = (exp((1)/(2)*z +((1)/(2)+ mu)*Pi*I))/(2*Pi*I*GAMMA((1)/(2)+ mu - kappa))* int((GAMMA(t - kappa)*GAMMA((1)/(2)+ mu - t))/(GAMMA((1)/(2)+ mu + t))*(z)^(t), t = - I*infinity..I*infinity)

Divide[1,Gamma[1 + 2*\[Mu]]]*WhittakerM[\[Kappa], \[Mu], Exp[+ Pi*I]*z] == Divide[Exp[Divide[1,2]*z +(Divide[1,2]+ \[Mu])*Pi*I],2*Pi*I*Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]]* Integrate[Divide[Gamma[t - \[Kappa]]*Gamma[Divide[1,2]+ \[Mu]- t],Gamma[Divide[1,2]+ \[Mu]+ t]]*(z)^(t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
13.16.E10 1 Γ ⁡ ( 1 + 2 ⁢ μ ) ⁢ M κ , μ ⁡ ( e - π ⁢ i ⁢ z ) = e 1 2 ⁢ z - ( 1 2 + μ ) ⁢ π ⁢ i 2 ⁢ π ⁢ i ⁢ Γ ⁡ ( 1 2 + μ - κ ) ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ Γ ⁡ ( t - κ ) ⁢ Γ ⁡ ( 1 2 + μ - t ) Γ ⁡ ( 1 2 + μ + t ) ⁢ z t ⁢ d t 1 Euler-Gamma 1 2 𝜇 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 superscript 𝑒 𝜋 imaginary-unit 𝑧 superscript 𝑒 1 2 𝑧 1 2 𝜇 𝜋 imaginary-unit 2 𝜋 imaginary-unit Euler-Gamma 1 2 𝜇 𝜅 superscript subscript imaginary-unit imaginary-unit Euler-Gamma 𝑡 𝜅 Euler-Gamma 1 2 𝜇 𝑡 Euler-Gamma 1 2 𝜇 𝑡 superscript 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle\frac{1}{\Gamma\left(1+2\mu\right)}M_{\kappa,\mu}% \left(e^{-\pi\mathrm{i}}z\right)=\frac{e^{\frac{1}{2}z-(\frac{1}{2}+\mu)\pi% \mathrm{i}}}{2\pi\mathrm{i}\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}\*\int_{-% \mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(t-\kappa\right)\Gamma% \left(\frac{1}{2}+\mu-t\right)}{\Gamma\left(\frac{1}{2}+\mu+t\right)}z^{t}% \mathrm{d}t}}
\frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{e^{-\pi\iunit}z} = \frac{e^{\frac{1}{2}z-(\frac{1}{2}+\mu)\pi\iunit}}{2\pi\iunit\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{t-\kappa}\EulerGamma@{\frac{1}{2}+\mu-t}}{\EulerGamma@{\frac{1}{2}+\mu+t}}z^{t}\diff{t}		 | ph ⁡ z | < 1 2 ⁢ π , ℜ ⁡ ( 1 + 2 ⁢ μ ) > 0 , ℜ ⁡ ( 1 2 + μ - κ ) > 0 , ℜ ⁡ ( t - κ ) > 0 , ℜ ⁡ ( 1 2 + μ - t ) > 0 , ℜ ⁡ ( 1 2 + μ + t ) > 0 formulae-sequence phase 𝑧 1 2 𝜋 formulae-sequence 1 2 𝜇 0 formulae-sequence 1 2 𝜇 𝜅 0 formulae-sequence 𝑡 𝜅 0 formulae-sequence 1 2 𝜇 𝑡 0 1 2 𝜇 𝑡 0 {\displaystyle{\displaystyle|\operatorname{ph}{z}|<\tfrac{1}{2}\pi,\Re(1+2\mu)% >0,\Re(\frac{1}{2}+\mu-\kappa)>0,\Re(t-\kappa)>0,\Re(\frac{1}{2}+\mu-t)>0,\Re(% \frac{1}{2}+\mu+t)>0}} 
(1)/(GAMMA(1 + 2*mu))*WhittakerM(kappa, mu, exp(- Pi*I)*z) = (exp((1)/(2)*z -((1)/(2)+ mu)*Pi*I))/(2*Pi*I*GAMMA((1)/(2)+ mu - kappa))* int((GAMMA(t - kappa)*GAMMA((1)/(2)+ mu - t))/(GAMMA((1)/(2)+ mu + t))*(z)^(t), t = - I*infinity..I*infinity)

Divide[1,Gamma[1 + 2*\[Mu]]]*WhittakerM[\[Kappa], \[Mu], Exp[- Pi*I]*z] == Divide[Exp[Divide[1,2]*z -(Divide[1,2]+ \[Mu])*Pi*I],2*Pi*I*Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]]* Integrate[Divide[Gamma[t - \[Kappa]]*Gamma[Divide[1,2]+ \[Mu]- t],Gamma[Divide[1,2]+ \[Mu]+ t]]*(z)^(t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
13.16.E11 W κ , μ ⁡ ( z ) = e - 1 2 ⁢ z 2 ⁢ π ⁢ i ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ Γ ⁡ ( 1 2 + μ + t ) ⁢ Γ ⁡ ( 1 2 - μ + t ) ⁢ Γ ⁡ ( - κ - t ) Γ ⁡ ( 1 2 + μ - κ ) ⁢ Γ ⁡ ( 1 2 - μ - κ ) ⁢ z - t ⁢ d t Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 superscript 𝑒 1 2 𝑧 2 𝜋 imaginary-unit superscript subscript imaginary-unit imaginary-unit Euler-Gamma 1 2 𝜇 𝑡 Euler-Gamma 1 2 𝜇 𝑡 Euler-Gamma 𝜅 𝑡 Euler-Gamma 1 2 𝜇 𝜅 Euler-Gamma 1 2 𝜇 𝜅 superscript 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle W_{\kappa,\mu}\left(z\right)=\frac{e^{-\frac{1}{2% }z}}{2\pi\mathrm{i}}\*\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma% \left(\frac{1}{2}+\mu+t\right)\Gamma\left(\frac{1}{2}-\mu+t\right)\Gamma\left(% -\kappa-t\right)}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1% }{2}-\mu-\kappa\right)}z^{-t}\mathrm{d}t}}
\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{e^{-\frac{1}{2}z}}{2\pi\iunit}\*\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{\frac{1}{2}+\mu+t}\EulerGamma@{\frac{1}{2}-\mu+t}\EulerGamma@{-\kappa-t}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}z^{-t}\diff{t}		 | ph ⁡ z | < 3 2 ⁢ π , ℜ ⁡ ( 1 2 + μ + t ) > 0 , ℜ ⁡ ( 1 2 - μ + t ) > 0 , ℜ ⁡ ( - κ - t ) > 0 , ℜ ⁡ ( 1 2 + μ - κ ) > 0 , ℜ ⁡ ( 1 2 - μ - κ ) > 0 formulae-sequence phase 𝑧 3 2 𝜋 formulae-sequence 1 2 𝜇 𝑡 0 formulae-sequence 1 2 𝜇 𝑡 0 formulae-sequence 𝜅 𝑡 0 formulae-sequence 1 2 𝜇 𝜅 0 1 2 𝜇 𝜅 0 {\displaystyle{\displaystyle|\operatorname{ph}{z}|<\tfrac{3}{2}\pi,\Re(\frac{1% }{2}+\mu+t)>0,\Re(\frac{1}{2}-\mu+t)>0,\Re(-\kappa-t)>0,\Re(\frac{1}{2}+\mu-% \kappa)>0,\Re(\frac{1}{2}-\mu-\kappa)>0}} 
WhittakerW(kappa, mu, z) = (exp(-(1)/(2)*z))/(2*Pi*I)* int((GAMMA((1)/(2)+ mu + t)*GAMMA((1)/(2)- mu + t)*GAMMA(- kappa - t))/(GAMMA((1)/(2)+ mu - kappa)*GAMMA((1)/(2)- mu - kappa))*(z)^(- t), t = - I*infinity..I*infinity)

WhittakerW[\[Kappa], \[Mu], z] == Divide[Exp[-Divide[1,2]*z],2*Pi*I]* Integrate[Divide[Gamma[Divide[1,2]+ \[Mu]+ t]*Gamma[Divide[1,2]- \[Mu]+ t]*Gamma[- \[Kappa]- t],Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]*Gamma[Divide[1,2]- \[Mu]- \[Kappa]]]*(z)^(- t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
13.16.E12 W κ , μ ⁡ ( z ) = e 1 2 ⁢ z 2 ⁢ π ⁢ i ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ Γ ⁡ ( 1 2 + μ + t ) ⁢ Γ ⁡ ( 1 2 - μ + t ) Γ ⁡ ( 1 - κ + t ) ⁢ z - t ⁢ d t Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 superscript 𝑒 1 2 𝑧 2 𝜋 imaginary-unit superscript subscript imaginary-unit imaginary-unit Euler-Gamma 1 2 𝜇 𝑡 Euler-Gamma 1 2 𝜇 𝑡 Euler-Gamma 1 𝜅 𝑡 superscript 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle W_{\kappa,\mu}\left(z\right)=\frac{e^{\frac{1}{2}% z}}{2\pi\mathrm{i}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma% \left(\frac{1}{2}+\mu+t\right)\Gamma\left(\frac{1}{2}-\mu+t\right)}{\Gamma% \left(1-\kappa+t\right)}z^{-t}\mathrm{d}t}}
\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{e^{\frac{1}{2}z}}{2\pi\iunit}\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{\frac{1}{2}+\mu+t}\EulerGamma@{\frac{1}{2}-\mu+t}}{\EulerGamma@{1-\kappa+t}}z^{-t}\diff{t}		 | ph ⁡ z | < 1 2 ⁢ π , ℜ ⁡ ( 1 2 + μ + t ) > 0 , ℜ ⁡ ( 1 2 - μ + t ) > 0 , ℜ ⁡ ( 1 - κ + t ) > 0 formulae-sequence phase 𝑧 1 2 𝜋 formulae-sequence 1 2 𝜇 𝑡 0 formulae-sequence 1 2 𝜇 𝑡 0 1 𝜅 𝑡 0 {\displaystyle{\displaystyle|\operatorname{ph}{z}|<\tfrac{1}{2}\pi,\Re(\frac{1% }{2}+\mu+t)>0,\Re(\frac{1}{2}-\mu+t)>0,\Re(1-\kappa+t)>0}} 
WhittakerW(kappa, mu, z) = (exp((1)/(2)*z))/(2*Pi*I)*int((GAMMA((1)/(2)+ mu + t)*GAMMA((1)/(2)- mu + t))/(GAMMA(1 - kappa + t))*(z)^(- t), t = - I*infinity..I*infinity)

WhittakerW[\[Kappa], \[Mu], z] == Divide[Exp[Divide[1,2]*z],2*Pi*I]*Integrate[Divide[Gamma[Divide[1,2]+ \[Mu]+ t]*Gamma[Divide[1,2]- \[Mu]+ t],Gamma[1 - \[Kappa]+ t]]*(z)^(- t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
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