# §12.12 Integrals

 12.12.1 $\int_{0}^{\infty}e^{-\frac{1}{4}t^{2}}t^{\mu-1}U\left(a,t\right)\mathrm{d}t=% \frac{\sqrt{\pi}2^{-\frac{1}{2}(\mu+a+\frac{1}{2})}\Gamma\left(\mu\right)}{% \Gamma\left(\frac{1}{2}(\mu+a+\frac{3}{2})\right)},$ $\Re\mu>0$ ,
 12.12.2 $\int_{0}^{\infty}e^{-\frac{3}{4}t^{2}}t^{-a-\frac{3}{2}}U\left(a,t\right)% \mathrm{d}t=2^{\frac{1}{4}+\frac{1}{2}a}\Gamma\left(-a-\tfrac{1}{2}\right)\cos% \left((\tfrac{1}{4}a+\tfrac{1}{8})\pi\right),$ $\Re a<-\tfrac{1}{2}$,
 12.12.3 $\int_{0}^{\infty}e^{-\frac{1}{4}t^{2}}t^{-a-\frac{1}{2}}(x^{2}+t^{2})^{-1}U% \left(a,t\right)\mathrm{d}t=\sqrt{\pi/2}\Gamma\left(\tfrac{1}{2}-a\right)x^{-a% -\frac{3}{2}}e^{\frac{1}{4}x^{2}}U\left(-a,x\right),$ $\Re a<\tfrac{1}{2},x>0$.

## Nicholson-type Integral

 12.12.4 $(U\left(a,z\right))^{2}+(\overline{U}\left(a,z\right))^{2}=\frac{2^{\frac{3}{2% }}}{\pi}\Gamma\left(\tfrac{1}{2}-a\right)\int_{0}^{\infty}\frac{e^{2at+\frac{1% }{2}z^{2}\tanh t}}{\sqrt{\sinh(2t)}}\mathrm{d}t,$ $\Re a<\tfrac{1}{2}$ .

When $z$ $(=x)$ is real the left-hand side equals $(F(a,x))^{2}$; compare (12.2.22).

For further integrals see §§13.10, 13.23, and use (12.7.14).

For compendia of integrals see Erdélyi et al. (1953b, v. 2, pp. 121–122), Erdélyi et al. (1954a, b, v. 1, pp. 60–61, 115, 210–211, and 336; v. 2, pp. 76–80, 115, 151, 171, and 395–398), Gradshteyn and Ryzhik (2000, §7.7), Magnus et al. (1966, pp. 330–331), Marichev (1983, pp. 190–191), Oberhettinger (1974, pp. 144–145), Oberhettinger (1990, pp. 106–108 and 192), Oberhettinger and Badii (1973, pp. 181–185), Prudnikov et al. (1986b, pp. 36–37, 155–168, 243–246, 289–290, 327–328, 419–420, and 619), Prudnikov et al. (1992a, §3.11), and Prudnikov et al. (1992b, §3.11).