§12.5 Integral Representations

§12.5(i) Integrals Along the Real Line

 12.5.1 $\mathop{U\/}\nolimits\!\left(a,z\right)=\frac{e^{-\frac{1}{4}z^{2}}}{\mathop{% \Gamma\/}\nolimits\!\left(\frac{1}{2}+a\right)}\int_{0}^{\infty}t^{a-\frac{1}{% 2}}e^{-\frac{1}{2}t^{2}-zt}dt,$ $\realpart{a}>-\tfrac{1}{2}$ ,
 12.5.2 $\mathop{U\/}\nolimits\!\left(a,z\right)=\frac{ze^{-\frac{1}{4}z^{2}}}{\mathop{% \Gamma\/}\nolimits\!\left(\frac{1}{4}+\frac{1}{2}a\right)}\*\int_{0}^{\infty}t% ^{\frac{1}{2}a-\frac{3}{4}}e^{-t}\left(z^{2}+2t\right)^{-\frac{1}{2}a-\frac{3}% {4}}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$, $\realpart{a}>-\tfrac{1}{2}$ ,
 12.5.3 $\mathop{U\/}\nolimits\!\left(a,z\right)=\frac{e^{-\frac{1}{4}z^{2}}}{\mathop{% \Gamma\/}\nolimits\!\left(\frac{3}{4}+\frac{1}{2}a\right)}\*\int_{0}^{\infty}t% ^{\frac{1}{2}a-\frac{1}{4}}e^{-t}\left(z^{2}+2t\right)^{-\frac{1}{2}a-\frac{1}% {4}}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$, $\realpart{a}>-\tfrac{3}{2}$ ,
 12.5.4 $\mathop{U\/}\nolimits\!\left(a,z\right)=\sqrt{\frac{2}{\pi}}e^{\frac{1}{4}z^{2% }}\*\int_{0}^{\infty}t^{-a-\frac{1}{2}}e^{-\frac{1}{2}t^{2}}\mathop{\cos\/}% \nolimits\!\left(zt+\left(\tfrac{1}{2}a+\tfrac{1}{4}\right)\pi\right)dt,$ $\realpart{a}<\tfrac{1}{2}$ .

§12.5(ii) Contour Integrals

The following integrals correspond to those of §12.5(i).

 12.5.5 $\mathop{U\/}\nolimits\!\left(a,z\right)=\frac{\mathop{\Gamma\/}\nolimits\!% \left(\frac{1}{2}-a\right)}{2\pi i}e^{-\frac{1}{4}z^{2}}\int_{-\infty}^{(0+)}e% ^{zt-\frac{1}{2}t^{2}}t^{a-\frac{1}{2}}dt,$ $a\neq\frac{1}{2},\frac{3}{2},\frac{5}{2},\dots$, $-\pi<\mathop{\mathrm{ph}\/}\nolimits t<\pi$.

Restrictions on $a$ are not needed in the following two representations:

 12.5.6 $\mathop{U\/}\nolimits\!\left(a,z\right)=\frac{e^{\frac{1}{4}z^{2}}}{i\sqrt{2% \pi}}\int_{c-i\infty}^{c+i\infty}e^{-zt+\frac{1}{2}t^{2}}t^{-a-\frac{1}{2}}dt,$ $-\tfrac{1}{2}\pi<\mathop{\mathrm{ph}\/}\nolimits t<\tfrac{1}{2}\pi$, $c>0$ ,
 12.5.7 $\mathop{V\/}\nolimits\!\left(a,z\right)=\frac{e^{-\frac{1}{4}z^{2}}}{2\pi}\*% \left(\int_{-ic-\infty}^{-ic+\infty}+\int_{ic-\infty}^{ic+\infty}\right)e^{zt-% \frac{1}{2}t^{2}}t^{a-\frac{1}{2}}dt,$ $-\pi<\mathop{\mathrm{ph}\/}\nolimits{t}<\pi$, $c>0$.

For proofs and further results see Miller (1955, §4) and Whittaker (1902).

§12.5(iii) Mellin–Barnes Integrals

 12.5.8 $\displaystyle\mathop{U\/}\nolimits\!\left(a,z\right)$ $\displaystyle=\frac{e^{-\frac{1}{4}z^{2}}z^{-a-\frac{1}{2}}}{2\pi i\mathop{% \Gamma\/}\nolimits\!\left(\frac{1}{2}+a\right)}\*\int_{-i\infty}^{i\infty}% \mathop{\Gamma\/}\nolimits\!\left(t\right)\mathop{\Gamma\/}\nolimits\!\left(% \tfrac{1}{2}+a-2t\right)2^{t}z^{2t}dt,$ $a\neq-\frac{1}{2},-\frac{3}{2},-\frac{5}{2},\dots$, $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{3}{4}\pi$, where the contour separates the poles of $\mathop{\Gamma\/}\nolimits\!\left(t\right)$ from those of $\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}+a-2t\right)$. 12.5.9 $\displaystyle\mathop{V\/}\nolimits\!\left(a,z\right)$ $\displaystyle=\sqrt{\frac{2}{\pi}}\frac{e^{\frac{1}{4}z^{2}}z^{a-\frac{1}{2}}}% {2\pi i\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-a\right)}\*\int_{-i\infty% }^{i\infty}\mathop{\Gamma\/}\nolimits\!\left(t\right)\mathop{\Gamma\/}% \nolimits\!\left(\tfrac{1}{2}-a-2t\right)2^{t}z^{2t}\mathop{\cos\/}\nolimits(% \pi t)dt,$ $a\neq\frac{1}{2},\frac{3}{2},\frac{5}{2},\dots$, $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{4}\pi$,

where the contour separates the poles of $\mathop{\Gamma\/}\nolimits\!\left(t\right)$ from those of $\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}-a-2t\right)$.

§12.5(iv) Compendia

For further collections of integral representations see Apelblat (1983, pp. 427-436), Erdélyi et al. (1953b, v. 2, pp. 119–120), Erdélyi et al. (1954a, pp. 289–291 and 362), Gradshteyn and Ryzhik (2000, §§9.24–9.25), Magnus et al. (1966, pp. 328–330), Oberhettinger (1974, pp. 251–252), and Oberhettinger and Badii (1973, pp. 378–384).