# §13.10 Integrals

## §13.10(i) Indefinite Integrals

When $a\neq 1$,

 13.10.1 $\int{\mathbf{M}}\left(a,b,z\right)\mathrm{d}z=\frac{1}{a-1}{\mathbf{M}}\left(a% -1,b-1,z\right),$
 13.10.2 $\int U\left(a,b,z\right)\mathrm{d}z=-\frac{1}{a-1}U\left(a-1,b-1,z\right).$

Other formulas of this kind can be constructed by inversion of the differentiation formulas given in §13.3(ii).

## §13.10(ii) Laplace Transforms

For the notation see §§15.1, 15.2(i), and 10.25(ii).

 13.10.3 $\int_{0}^{\infty}e^{-zt}t^{b-1}{\mathbf{M}}\left(a,c,kt\right)\mathrm{d}t=% \Gamma\left(b\right)z^{-b}{{}_{2}{\mathbf{F}}_{1}}\left(a,b;c;\ifrac{k}{z}% \right),$ $\Re b>0$, $\Re z>\max\left(\Re k,0\right)$,
 13.10.4 $\int_{0}^{\infty}e^{-zt}t^{b-1}{\mathbf{M}}\left(a,b,t\right)\mathrm{d}t=z^{-b% }\left(1-\frac{1}{z}\right)^{-a},$ $\Re b>0$, $\Re z>1$,
 13.10.5 $\int_{0}^{\infty}e^{-t}t^{b-1}{\mathbf{M}}\left(a,c,t\right)\mathrm{d}t=\frac{% \Gamma\left(b\right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a\right)\Gamma% \left(c-b\right)},$ $\Re(c-a)>\Re b>0$,
 13.10.6 $\int_{0}^{\infty}e^{-zt-t^{2}}t^{2b-2}{\mathbf{M}}\left(a,b,t^{2}\right)% \mathrm{d}t=\tfrac{1}{2}\pi^{-\frac{1}{2}}\Gamma\left(b-\tfrac{1}{2}\right)U% \left(b-\tfrac{1}{2},a+\tfrac{1}{2},\tfrac{1}{4}z^{2}\right),$ $\Re b>\tfrac{1}{2}$, $\Re z>0$,
 13.10.7 $\int_{0}^{\infty}e^{-zt}t^{b-1}U\left(a,c,t\right)\mathrm{d}t=\Gamma\left(b% \right)\Gamma\left(b-c+1\right)\*z^{-b}{{}_{2}{\mathbf{F}}_{1}}\left(a,b;a+b-c% +1;1-\frac{1}{z}\right),$ $\Re b>\max\left(\Re c-1,0\right)$, $\Re z>0$.

### Loop Integrals

 13.10.8 $\frac{1}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{tz}t^{-a}{\mathbf{M}}\left(a,b% ,\ifrac{y}{t}\right)\mathrm{d}t=\frac{1}{\Gamma\left(a\right)}z^{\frac{1}{2}(2% a-b-1)}y^{\frac{1}{2}(1-b)}I_{b-1}\left(2\sqrt{zy}\right),$ $\Re z>0$.
 13.10.9 $\frac{1}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{tz}t^{-a}U\left(a,b,\ifrac{y}{% t}\right)\mathrm{d}t=\frac{2z^{\frac{1}{2}(2a-b-1)}y^{\frac{1}{2}(1-b)}}{% \Gamma\left(a\right)\Gamma\left(a-b+1\right)}K_{b-1}\left(2\sqrt{zy}\right),$ $\Re z>0$.

For additional Laplace transforms see Erdélyi et al. (1954a, §§4.22, 5.20), Oberhettinger and Badii (1973, §1.17), and Prudnikov et al. (1992a, §§3.34, 3.35). Inverse Laplace transforms are given in Oberhettinger and Badii (1973, §2.16) and Prudnikov et al. (1992b, §§3.33, 3.34).

## §13.10(iii) Mellin Transforms

 13.10.10 $\int_{0}^{\infty}t^{\lambda-1}{\mathbf{M}}\left(a,b,-t\right)\mathrm{d}t=\frac% {\Gamma\left(\lambda\right)\Gamma\left(a-\lambda\right)}{\Gamma\left(a\right)% \Gamma\left(b-\lambda\right)},$ $0<\Re\lambda<\Re a$,
 13.10.11 $\int_{0}^{\infty}t^{\lambda-1}U\left(a,b,t\right)\mathrm{d}t=\frac{\Gamma\left% (\lambda\right)\Gamma\left(a-\lambda\right)\Gamma\left(\lambda-b+1\right)}{% \Gamma\left(a\right)\Gamma\left(a-b+1\right)},$ $\max\left(\Re b-1,0\right)<\Re\lambda<\Re a$.

For additional Mellin transforms see Erdélyi et al. (1954a, §§6.9, 7.5), Marichev (1983, pp. 283–287), and Oberhettinger (1974, §§1.13, 2.8).

## §13.10(iv) Fourier Transforms

 13.10.12 $\int_{0}^{\infty}\cos\left(2xt\right){\mathbf{M}}\left(a,b,-t^{2}\right)% \mathrm{d}t=\frac{\sqrt{\pi}}{2\Gamma\left(a\right)}x^{2a-1}e^{-x^{2}}U\left(b% -\tfrac{1}{2},a+\tfrac{1}{2},x^{2}\right),$ $\Re a>0$.

For additional Fourier transforms see Erdélyi et al. (1954a, §§1.14, 2.14, 3.3) and Oberhettinger (1990, §§1.22, 2.22).

## §13.10(v) Hankel Transforms

For the notation see §10.2(ii).

 13.10.13 $\int_{0}^{\infty}e^{-t}t^{b-1-\frac{1}{2}\nu}{\mathbf{M}}\left(a,b,t\right)J_{% \nu}\left(2\sqrt{xt}\right)\mathrm{d}t=x^{-a+\frac{1}{2}\nu}e^{-x}{\mathbf{M}}% \left(\nu-b+1,\nu-a+1,x\right),$ $x>0$, $2\Re a<\Re\nu+\tfrac{5}{2}$, $\Re b>0$,
 13.10.14 $\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}{\mathbf{M}}\left(a,b,t\right)J_{\nu}% \left(2\sqrt{xt}\right)\mathrm{d}t=\frac{x^{\frac{1}{2}\nu}e^{-x}}{\Gamma\left% (b-a\right)}U\left(a,a-b+\nu+2,x\right),$ $x>0$, $-1<\Re\nu<2\Re(b-a)-\tfrac{1}{2}$,
 13.10.15 $\int_{0}^{\infty}t^{\frac{1}{2}\nu}U\left(a,b,t\right)J_{\nu}\left(2\sqrt{xt}% \right)\mathrm{d}t=\frac{\Gamma\left(\nu-b+2\right)}{\Gamma\left(a\right)}x^{% \frac{1}{2}\nu}U\left(\nu-b+2,\nu-a+2,x\right),$ $x>0$, $\max\left(\Re b-2,-1\right)<\Re\nu<2\Re a+\tfrac{1}{2}$,
 13.10.16 $\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}U\left(a,b,t\right)J_{\nu}\left(2% \sqrt{xt}\right)\mathrm{d}t=\Gamma\left(\nu-b+2\right)x^{\frac{1}{2}\nu}e^{-x}% {\mathbf{M}}\left(a,a-b+\nu+2,x\right),$ $x>0$, $\max\left(\Re b-2,-1\right)<\Re\nu$.

For additional Hankel transforms and also other Bessel transforms see Erdélyi et al. (1954b, §8.18) and Oberhettinger (1972, §§1.16 and 3.4.42–46, 4.4.45–47, 5.94–97).

## §13.10(vi) Other Integrals

For integral transforms in terms of Whittaker functions see §13.23(iv). Additional integrals can be found in Apelblat (1983, pp. 388–392), Erdélyi et al. (1954b), Gradshteyn and Ryzhik (2000, §7.6), Magnus et al. (1966, §6.1.2), Prudnikov et al. (1990, §§1.13, 1.14, 2.19, 4.2.2), Prudnikov et al. (1992a, §§3.35, 3.36), and Prudnikov et al. (1992b, §§3.33, 3.34). See also (13.4.2), (13.4.5), (13.4.6).

Generalized orthogonality integrals (33.14.13) and (33.14.15) can be expressed in terms of Kummer functions via the definitions in that section.