# §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 $\mathop{J_{2\mu}\/}\nolimits$, $\mathop{I_{2\mu}\/}\nolimits$, and $\mathop{K_{2\mu}\/}\nolimits$, and §§15.1, 15.2(i) for $\mathop{{{}_{2}{\mathbf{F}}_{1}}\/}\nolimits$.

 13.16.1 $\displaystyle\mathop{M_{\kappa,\mu}\/}\nolimits\!\left(z\right)$ $\displaystyle=\frac{\mathop{\Gamma\/}\nolimits\!\left(1+2\mu\right)z^{\mu+% \frac{1}{2}}2^{-2\mu}}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu-% \kappa\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu+\kappa\right)}% \*\int_{-1}^{1}e^{\frac{1}{2}zt}(1+t)^{\mu-\frac{1}{2}-\kappa}(1-t)^{\mu-\frac% {1}{2}+\kappa}\mathrm{d}t,$ $\Re{\mu}+\tfrac{1}{2}>\left|\Re{\kappa}\right|$, 13.16.2 $\displaystyle\mathop{M_{\kappa,\mu}\/}\nolimits\!\left(z\right)$ $\displaystyle=\frac{\mathop{\Gamma\/}\nolimits\!\left(1+2\mu\right)z^{\lambda}% }{\mathop{\Gamma\/}\nolimits\!\left(1+2\mu-2\lambda\right)\mathop{\Gamma\/}% \nolimits\!\left(2\lambda\right)}\*\int_{0}^{1}\mathop{M_{\kappa-\lambda,\mu-% \lambda}\/}\nolimits\!\left(zt\right)e^{\frac{1}{2}z(t-1)}t^{\mu-\lambda-\frac% {1}{2}}{(1-t)^{2\lambda-1}}\mathrm{d}t,$ $\Re{\mu}+\tfrac{1}{2}>\Re{\lambda}>0$,
 13.16.3 $\displaystyle\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(1+2\mu\right)}\mathop{% M_{\kappa,\mu}\/}\nolimits\!\left(z\right)$ $\displaystyle=\frac{\sqrt{z}e^{\frac{1}{2}z}}{\mathop{\Gamma\/}\nolimits\!% \left(\frac{1}{2}+\mu+\kappa\right)}\int_{0}^{\infty}e^{-t}t^{\kappa-\frac{1}{% 2}}\mathop{J_{2\mu}\/}\nolimits\!\left(2\sqrt{zt}\right)\mathrm{d}t,$ $\Re{(\kappa+\mu)+\tfrac{1}{2}}>0$, 13.16.4 $\displaystyle\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(1+2\mu\right)}\mathop{% M_{\kappa,\mu}\/}\nolimits\!\left(z\right)$ $\displaystyle=\frac{\sqrt{z}e^{-\frac{1}{2}z}}{\mathop{\Gamma\/}\nolimits\!% \left(\frac{1}{2}+\mu-\kappa\right)}\*\int_{0}^{\infty}e^{-t}t^{-\kappa-\frac{% 1}{2}}\mathop{I_{2\mu}\/}\nolimits\!\left(2\sqrt{zt}\right)\mathrm{d}t,$ $\Re{(\kappa-\mu)-\tfrac{1}{2}}<0$. Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathop{M_{\NVar{\kappa},\NVar{\mu}}\/}\nolimits\!\left(\NVar{z}\right)$: Whittaker confluent hypergeometric function, $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of exponential function, $\int$: integral, $\mathop{I_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: modified Bessel function of the first kind, $\Re{}$: real part and $z$: complex variable Referenced by: 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 $\Re{(\kappa-\mu)-\tfrac{1}{2}}<0$. Reported 2012-07-30 See also: Annotations for 13.16(i)
 13.16.5 $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right)=\frac{z^{\mu+\frac{1}{2}}2^% {-2\mu}}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu-\kappa\right)}\*% \int_{1}^{\infty}e^{-\frac{1}{2}zt}(t-1)^{\mu-\frac{1}{2}-\kappa}(t+1)^{\mu-% \frac{1}{2}+\kappa}\mathrm{d}t,$ $\Re{\mu}+\tfrac{1}{2}>\Re{\kappa}$, $|\mathop{\mathrm{ph}\/}\nolimits{z}|<\frac{1}{2}\pi$,
 13.16.6 $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right)=\frac{e^{-\frac{1}{2}z}z^{% \kappa+1}}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu-\kappa\right)% \mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\mu-\kappa\right)}\*\int_{0}^{% \infty}\frac{\mathop{W_{-\kappa,\mu}\/}\nolimits\!\left(t\right)e^{-\frac{1}{2% }t}t^{-\kappa-1}}{t+z}\mathrm{d}t,$ $|\mathop{\mathrm{ph}\/}\nolimits{z}|<\pi$, $\Re{(\frac{1}{2}+\mu-\kappa)}>\max\left(2\Re{\mu},0\right)$,
 13.16.7 $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right)=\frac{(-1)^{n}e^{-\frac{1}{% 2}z}z^{\frac{1}{2}-\mu-n}}{\mathop{\Gamma\/}\nolimits\!\left(1+2\mu\right)% \mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\mu-\kappa\right)}\*\int_{0}^{% \infty}\frac{\mathop{M_{-\kappa,\mu}\/}\nolimits\!\left(t\right)e^{-\frac{1}{2% }t}t^{n+\mu-\frac{1}{2}}}{t+z}\mathrm{d}t,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$, $n=0,1,2,\dots$, $-\Re{(1+2\mu)},
 13.16.8 $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right)=\frac{2\sqrt{z}e^{-\frac{1}% {2}z}}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu-\kappa\right)\mathop{% \Gamma\/}\nolimits\!\left(\frac{1}{2}-\mu-\kappa\right)}\*\int_{0}^{\infty}e^{% -t}t^{-\kappa-\frac{1}{2}}\mathop{K_{2\mu}\/}\nolimits\!\left(2\sqrt{zt}\right% )\mathrm{d}t,$ $\Re{(\mu-\kappa)+\tfrac{1}{2}}>0$,
 13.16.9 $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right)=e^{-\frac{1}{2}z}z^{\kappa+% c}\*\int_{0}^{\infty}e^{-zt}t^{c-1}\mathop{{{}_{2}{\mathbf{F}}_{1}}\/}% \nolimits\!\left({\tfrac{1}{2}+\mu-\kappa,\tfrac{1}{2}-\mu-\kappa\atop c};-t% \right)\mathrm{d}t,$ $|\mathop{\mathrm{ph}\/}\nolimits{z}|<\frac{1}{2}\pi$,

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

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

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

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

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

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

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