# §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.