§13.16 Integral Representations

§13.16(i) Integrals Along the Real Line

In this subsection see §§10.2(ii), 10.25(ii) for the functions $J_{2\mu}$, $I_{2\mu}$, and $K_{2\mu}$, and §§15.1, 15.2(i) for ${{}_{2}{\mathbf{F}}_{1}}$.

 13.16.1 $\displaystyle M_{\kappa,\mu}\left(z\right)$ $\displaystyle=\frac{\Gamma\left(1+2\mu\right)z^{\mu+\frac{1}{2}}2^{-2\mu}}{% \Gamma\left(\frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}{2}+\mu+\kappa% \right)}\*\int_{-1}^{1}e^{\frac{1}{2}zt}(1+t)^{\mu-\frac{1}{2}-\kappa}(1-t)^{% \mu-\frac{1}{2}+\kappa}\mathrm{d}t,$ $\Re\mu+\tfrac{1}{2}>\left|\Re\kappa\right|$, 13.16.2 $\displaystyle M_{\kappa,\mu}\left(z\right)$ $\displaystyle=\frac{\Gamma\left(1+2\mu\right)z^{\lambda}}{\Gamma\left(1+2\mu-2% \lambda\right)\Gamma\left(2\lambda\right)}\*\int_{0}^{1}M_{\kappa-\lambda,\mu-% \lambda}\left(zt\right)e^{\frac{1}{2}z(t-1)}t^{\mu-\lambda-\frac{1}{2}}{(1-t)^% {2\lambda-1}}\mathrm{d}t,$ $\Re\mu+\tfrac{1}{2}>\Re\lambda>0$,
 13.16.3 $\displaystyle\frac{1}{\Gamma\left(1+2\mu\right)}M_{\kappa,\mu}\left(z\right)$ $\displaystyle=\frac{\sqrt{z}e^{\frac{1}{2}z}}{\Gamma\left(\frac{1}{2}+\mu+% \kappa\right)}\int_{0}^{\infty}e^{-t}t^{\kappa-\frac{1}{2}}J_{2\mu}\left(2% \sqrt{zt}\right)\mathrm{d}t,$ $\Re\left(\kappa+\mu\right)+\tfrac{1}{2}>0$, 13.16.4 $\displaystyle\frac{1}{\Gamma\left(1+2\mu\right)}M_{\kappa,\mu}\left(z\right)$ $\displaystyle=\frac{\sqrt{z}e^{-\frac{1}{2}z}}{\Gamma\left(\frac{1}{2}+\mu-% \kappa\right)}\*\int_{0}^{\infty}e^{-t}t^{-\kappa-\frac{1}{2}}I_{2\mu}\left(2% \sqrt{zt}\right)\mathrm{d}t,$ $\Re(\kappa-\mu)-\tfrac{1}{2}<0$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$: Whittaker confluent hypergeometric function, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the first kind, $\Re$: real part and $z$: complex variable Referenced by: Erratum (V1.0.5) for Equation (13.16.4) Permalink: http://dlmf.nist.gov/13.16.E4 Encodings: TeX, pMML, png Errata (effective with 1.0.5): Originally the condition for the validity of this equation was stated incorrectly as $\Re(\kappa-\mu)-\tfrac{1}{2}>0$. The correct condition is $\Re(\kappa-\mu)-\tfrac{1}{2}<0$. See also: Annotations for §13.16(i), §13.16 and Ch.13
 13.16.5 $W_{\kappa,\mu}\left(z\right)=\frac{z^{\mu+\frac{1}{2}}2^{-2\mu}}{\Gamma\left(% \frac{1}{2}+\mu-\kappa\right)}\*\int_{1}^{\infty}e^{-\frac{1}{2}zt}(t-1)^{\mu-% \frac{1}{2}-\kappa}(t+1)^{\mu-\frac{1}{2}+\kappa}\mathrm{d}t,$ $\Re\mu+\tfrac{1}{2}>\Re\kappa$, $|\operatorname{ph}{z}|<\frac{1}{2}\pi$,
 13.16.6 $W_{\kappa,\mu}\left(z\right)=\frac{e^{-\frac{1}{2}z}z^{\kappa+1}}{\Gamma\left(% \frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}{2}-\mu-\kappa\right)}\*\int_% {0}^{\infty}\frac{W_{-\kappa,\mu}\left(t\right)e^{-\frac{1}{2}t}t^{-\kappa-1}}% {t+z}\mathrm{d}t,$ $|\operatorname{ph}{z}|<\pi$, $\Re\left(\frac{1}{2}+\mu-\kappa\right)>\max\left(2\Re\mu,0\right)$,
 13.16.7 $W_{\kappa,\mu}\left(z\right)=\frac{(-1)^{n}e^{-\frac{1}{2}z}z^{\frac{1}{2}-\mu% -n}}{\Gamma\left(1+2\mu\right)\Gamma\left(\frac{1}{2}-\mu-\kappa\right)}\*\int% _{0}^{\infty}\frac{M_{-\kappa,\mu}\left(t\right)e^{-\frac{1}{2}t}t^{n+\mu-% \frac{1}{2}}}{t+z}\mathrm{d}t,$ $|\operatorname{ph}z|<\pi$, $n=0,1,2,\dots$, $-\Re\left(1+2\mu\right),
 13.16.8 $W_{\kappa,\mu}\left(z\right)=\frac{2\sqrt{z}e^{-\frac{1}{2}z}}{\Gamma\left(% \frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}{2}-\mu-\kappa\right)}\*\int_% {0}^{\infty}e^{-t}t^{-\kappa-\frac{1}{2}}K_{2\mu}\left(2\sqrt{zt}\right)% \mathrm{d}t,$ $\Re\left(\mu-\kappa\right)+\tfrac{1}{2}>0$,
 13.16.9 $W_{\kappa,\mu}\left(z\right)=e^{-\frac{1}{2}z}z^{\kappa+c}\*\int_{0}^{\infty}e% ^{-zt}t^{c-1}{{}_{2}{\mathbf{F}}_{1}}\left({\tfrac{1}{2}+\mu-\kappa,\tfrac{1}{% 2}-\mu-\kappa\atop c};-t\right)\mathrm{d}t,$ $|\operatorname{ph}{z}|<\frac{1}{2}\pi$,

where $c$ is arbitrary, $\Re c>0$.

§13.16(ii) Contour Integrals

For contour integral representations combine (13.14.2) and (13.14.3) with §13.4(ii). See Buchholz (1969, §2.3), Erdélyi et al. (1953a, §6.11.3), and Slater (1960, Chapter 3). See also §13.16(iii).

§13.16(iii) Mellin–Barnes Integrals

If $\tfrac{1}{2}+\mu-\kappa\neq 0,-1,-2,\dots$, then

 13.16.10 $\frac{1}{\Gamma\left(1+2\mu\right)}M_{\kappa,\mu}\left(e^{\pm\pi\mathrm{i}}z% \right)=\frac{e^{\frac{1}{2}z\pm(\frac{1}{2}+\mu)\pi\mathrm{i}}}{2\pi\mathrm{i% }\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}\*\int_{-\mathrm{i}\infty}^{\mathrm% {i}\infty}\frac{\Gamma\left(t-\kappa\right)\Gamma\left(\frac{1}{2}+\mu-t\right% )}{\Gamma\left(\frac{1}{2}+\mu+t\right)}z^{t}\mathrm{d}t,$ $|\operatorname{ph}{z}|<\tfrac{1}{2}\pi$,

where the contour of integration separates the poles of $\Gamma\left(t-\kappa\right)$ from those of $\Gamma\left(\frac{1}{2}+\mu-t\right)$.

If $\tfrac{1}{2}\pm\mu-\kappa\neq 0,-1,-2,\dots$, then

 13.16.11 $W_{\kappa,\mu}\left(z\right)=\frac{e^{-\frac{1}{2}z}}{2\pi\mathrm{i}}\*\int_{-% \mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(\frac{1}{2}+\mu+t\right)% \Gamma\left(\frac{1}{2}-\mu+t\right)\Gamma\left(-\kappa-t\right)}{\Gamma\left(% \frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}{2}-\mu-\kappa\right)}z^{-t}% \mathrm{d}t,$ $|\operatorname{ph}{z}|<\tfrac{3}{2}\pi$,

where the contour of integration separates the poles of $\Gamma\left(\frac{1}{2}+\mu+t\right)\Gamma\left(\frac{1}{2}-\mu+t\right)$ from those of $\Gamma\left(-\kappa-t\right)$.

 13.16.12 $W_{\kappa,\mu}\left(z\right)=\frac{e^{\frac{1}{2}z}}{2\pi\mathrm{i}}\int_{-% \mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(\frac{1}{2}+\mu+t\right)% \Gamma\left(\frac{1}{2}-\mu+t\right)}{\Gamma\left(1-\kappa+t\right)}z^{-t}% \mathrm{d}t,$ $|\operatorname{ph}{z}|<\tfrac{1}{2}\pi$,

where the contour of integration passes all the poles of $\Gamma\left(\frac{1}{2}+\mu+t\right)\Gamma\left(\frac{1}{2}-\mu+t\right)$ on the right-hand side.