15 Hypergeometric Function Properties 15.13 Zeros 15.15 Sums

§15.14 Integrals

Notes:: For (15.14.1) see Andrews et al. (1999, §2.4).
Keywords:: Mellin transform, hypergeometric function, integrals
Permalink:: http://dlmf.nist.gov/15.14

The Mellin transform of the hypergeometric function of negative argument is given by

15.14.1 $\int_{0}^{\infty}x^{{s-1}}\mathop{\mathbf{F}\/}\nolimits\!\left({a,b\atop c};-% x\right)dx=\frac{\mathop{\Gamma\/}\nolimits\!\left(s\right)\mathop{\Gamma\/}% \nolimits\!\left(a-s\right)\mathop{\Gamma\/}\nolimits\!\left(b-s\right)}{% \mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(b% \right)\mathop{\Gamma\/}\nolimits\!\left(c-s\right)},$ $\min(\realpart{a},\realpart{b})>\realpart{s}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , $\int$ : integral, : open interval, $\realpart{}$ : real part, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, : real variable, : nonnegative integer, : real or complex parameter, : real or complex parameter and : real or complex parameter
Referenced by:: §15.14
Permalink:: http://dlmf.nist.gov/15.14.E1
Encodings:: TeX, pMML, png

Integrals of the form $\int x^{\alpha}(x+t)^{\beta}\mathop{F\/}\nolimits\!\left(a,b;c;x\right)dx$ and more complicated forms are given in Apelblat (1983, pp. 370–387), Prudnikov et al. (1990, §§1.15 and 2.21), and Gradshteyn and Ryzhik (2000, §7.5).

Fourier transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §§1.14 and 2.14). Laplace transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §4.21), Oberhettinger and Badii (1973, §1.19), and Prudnikov et al. (1992a, §3.37). Inverse Laplace transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §5.19), Oberhettinger and Badii (1973, §2.18), and Prudnikov et al. (1992b, §3.35). Mellin transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §6.9), Oberhettinger (1974, §1.15), and Marichev (1983, pp. 288–299). Inverse Mellin transforms are given in Erdélyi et al. (1954a, §7.5). Hankel transforms of hypergeometric functions are given in Oberhettinger (1972, §1.17) and Erdélyi et al. (1954b, §8.17).

For other integral transforms see Erdélyi et al. (1954b), Prudnikov et al. (1992b, §4.3.43), and also §15.9(ii).