§13.23 Integrals

§13.23(i) Laplace and Mellin Transforms

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

 13.23.1 $\int_{0}^{\infty}e^{-zt}t^{\nu-1}M_{\kappa,\mu}\left(t\right)\mathrm{d}t=\frac% {\Gamma\left(\mu+\nu+\tfrac{1}{2}\right)}{\left(z+\frac{1}{2}\right)^{\mu+\nu+% \frac{1}{2}}}\*{{}_{2}F_{1}}\left({\tfrac{1}{2}+\mu-\kappa,\tfrac{1}{2}+\mu+% \nu\atop 1+2\mu};\frac{1}{z+\frac{1}{2}}\right),$ $\Re\mu+\nu+\tfrac{1}{2}>0$, $\Re z>\tfrac{1}{2}$.
 13.23.2 $\int_{0}^{\infty}e^{-zt}t^{\mu-\frac{1}{2}}M_{\kappa,\mu}\left(t\right)\mathrm% {d}t=\Gamma\left(2\mu+1\right)\left(z+\tfrac{1}{2}\right)^{-\kappa-\mu-\frac{1% }{2}}\*\left(z-\tfrac{1}{2}\right)^{\kappa-\mu-\frac{1}{2}},$ $\Re\mu>-\tfrac{1}{2}$, $\Re z>\tfrac{1}{2}$,
 13.23.3 $\frac{1}{\Gamma\left(1+2\mu\right)}\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\nu-1}% M_{\kappa,\mu}\left(t\right)\mathrm{d}t=\frac{\Gamma\left(\mu+\nu+\frac{1}{2}% \right)\Gamma\left(\kappa-\nu\right)}{\Gamma\left(\frac{1}{2}+\mu+\kappa\right% )\Gamma\left(\frac{1}{2}+\mu-\nu\right)},$ $-\tfrac{1}{2}-\Re\mu<\Re\nu<\Re\kappa$.
 13.23.4 $\int_{0}^{\infty}e^{-zt}t^{\nu-1}W_{\kappa,\mu}\left(t\right)\mathrm{d}t=% \Gamma\left(\tfrac{1}{2}+\mu+\nu\right)\Gamma\left(\tfrac{1}{2}-\mu+\nu\right)% \*{{}_{2}{\mathbf{F}}_{1}}\left({\tfrac{1}{2}-\mu+\nu,\tfrac{1}{2}+\mu+\nu% \atop\nu-\kappa+1};\tfrac{1}{2}-z\right),$ $\Re\left(\nu+\tfrac{1}{2}\right)>|\Re\mu|$, $\Re z>-\tfrac{1}{2}$,
 13.23.5 $\int_{0}^{\infty}e^{\frac{1}{2}t}t^{\nu-1}W_{\kappa,\mu}\left(t\right)\mathrm{% d}t=\frac{\Gamma\left(\frac{1}{2}+\mu+\nu\right)\Gamma\left(\frac{1}{2}-\mu+% \nu\right)\Gamma\left(-\kappa-\nu\right)}{\Gamma\left(\frac{1}{2}+\mu-\kappa% \right)\Gamma\left(\frac{1}{2}-\mu-\kappa\right)},$ $|\Re\mu|-\tfrac{1}{2}<\Re\nu<-\Re\kappa$.
 13.23.6 $\frac{1}{\Gamma\left(1+2\mu\right)2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{zt+% \frac{1}{2}t^{-1}}t^{\kappa}M_{\kappa,\mu}\left(t^{-1}\right)\mathrm{d}t=\frac% {z^{-\kappa-\frac{1}{2}}}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}I_{2\mu}% \left(2\sqrt{z}\right),$ $\Re z>0$.
 13.23.7 $\frac{1}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{zt+\frac{1}{2}t^{-1}}t^{\kappa% }W_{\kappa,\mu}\left(t^{-1}\right)\mathrm{d}t=\frac{2z^{-\kappa-\frac{1}{2}}}{% \Gamma\left(\frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}{2}-\mu-\kappa% \right)}K_{2\mu}\left(2\sqrt{z}\right),$ $\Re z>0$.

For additional Laplace and Mellin transforms see Erdélyi et al. (1954a, §§4.22, 5.20, 6.9, 7.5), Marichev (1983, pp. 283–287), Oberhettinger and Badii (1973, §1.17), Oberhettinger (1974, §§1.13, 2.8), 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).

§13.23(ii) Fourier Transforms

 13.23.8 $\frac{1}{\Gamma\left(1+2\mu\right)}\int_{0}^{\infty}\cos\left(2xt\right)e^{-% \frac{1}{2}t^{2}}t^{-2\mu-1}M_{\kappa,\mu}\left(t^{2}\right)\mathrm{d}t=\frac{% \sqrt{\pi}e^{-\frac{1}{2}x^{2}}x^{\mu+\kappa-1}}{2\Gamma\left(\frac{1}{2}+\mu+% \kappa\right)}W_{\frac{1}{2}\kappa-\frac{3}{2}\mu,\frac{1}{2}\kappa+\frac{1}{2% }\mu}\left(x^{2}\right),$ $\Re\left(\kappa+\mu\right)>-\tfrac{1}{2}$.

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

§13.23(iii) Hankel Transforms

For the notation see §10.2(ii).

 13.23.9 $\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\mu-\frac{1}{2}(\nu+1)}M_{\kappa,\mu}% \left(t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\frac{\Gamma\left(1+2% \mu\right)}{\Gamma\left(\frac{1}{2}-\mu+\kappa+\nu\right)}\*e^{-\frac{1}{2}x}x% ^{\frac{1}{2}(\kappa-\mu-\frac{3}{2})}\*M_{\frac{1}{2}(\kappa+3\mu-\nu+\frac{1% }{2}),\frac{1}{2}(\kappa-\mu+\nu-\frac{1}{2})}\left(x\right),$ $x>0$, $-\tfrac{1}{2}<\Re\mu<\Re\left(\kappa+\tfrac{1}{2}\nu\right)+\tfrac{3}{4}$,
 13.23.10 $\frac{1}{\Gamma\left(1+2\mu\right)}\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\frac{% 1}{2}(\nu-1)-\mu}M_{\kappa,\mu}\left(t\right)J_{\nu}\left(2\sqrt{xt}\right)% \mathrm{d}t=\frac{e^{-\frac{1}{2}x}x^{\frac{1}{2}(\kappa+\mu-\frac{3}{2})}}{% \Gamma\left(\frac{1}{2}+\mu+\kappa\right)}\*W_{\frac{1}{2}(\kappa-3\mu+\nu+% \frac{1}{2}),\frac{1}{2}(\kappa+\mu-\nu-\frac{1}{2})}\left(x\right),$ $x>0$, $-1<\Re\nu<2\Re\left(\mu+\kappa\right)+\tfrac{1}{2}$.
 13.23.11 $\int_{0}^{\infty}e^{\frac{1}{2}t}t^{\frac{1}{2}(\nu-1)-\mu}W_{\kappa,\mu}\left% (t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\frac{\Gamma\left(\nu-2\mu+% 1\right)}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}\*e^{\frac{1}{2}x}x^{\frac% {1}{2}(\mu-\kappa-\frac{3}{2})}\*W_{\frac{1}{2}(\kappa+3\mu-\nu-\frac{1}{2}),% \frac{1}{2}(\kappa-\mu+\nu+\frac{1}{2})}\left(x\right),$ $x>0$, $\max(2\Re\mu-1,-1)<\Re\nu<2\Re\mu-\kappa+\tfrac{3}{2}$,
 13.23.12 $\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\frac{1}{2}(\nu-1)-\mu}W_{\kappa,\mu}% \left(t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\frac{\Gamma\left(\nu-% 2\mu+1\right)}{\Gamma\left(\frac{3}{2}-\mu-\kappa+\nu\right)}\*e^{-\frac{1}{2}% x}x^{\frac{1}{2}(\mu+\kappa-\frac{3}{2})}\*M_{\frac{1}{2}(\kappa-3\mu+\nu+% \frac{1}{2}),\frac{1}{2}(\nu-\mu-\kappa+\frac{1}{2})}\left(x\right),$ $x>0$, $\max(2\Re\mu-1,-1)<\Re\nu$.

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).

§13.23(iv) Integral Transforms in terms of Whittaker Functions

Let $f(x)$ be absolutely integrable on the interval $[r,R]$ for all positive $r, $f(x)=O\left(x^{\rho_{0}}\right)$ as $x\to 0+$, and $f(x)=O\left(e^{-\rho_{1}x}\right)$ as $x\to+\infty$, where $\rho_{1}>\frac{1}{2}$. Then for $\mu$ in the half-plane $\Re\mu\geq\mu_{1}>\max\left(-\rho_{0},\Re\kappa-\frac{1}{2}\right)$

 13.23.13 $\displaystyle g(\mu)$ $\displaystyle=\frac{1}{\Gamma\left(1+2\mu\right)}\int_{0}^{\infty}f(x)x^{-% \frac{3}{2}}M_{\kappa,\mu}\left(x\right)\mathrm{d}x,$ 13.23.14 $\displaystyle f(x)$ $\displaystyle=\frac{1}{\pi\mathrm{i}\sqrt{x}}\int_{\mu_{1}-\mathrm{i}\infty}^{% \mu_{1}+\mathrm{i}\infty}\mu g(\mu)\Gamma\left(\tfrac{1}{2}+\mu-\kappa\right)W% _{\kappa,\mu}\left(x\right)\mathrm{d}\mu.$

For additional integral transforms see Magnus et al. (1966, p. 189), Prudnikov et al. (1992b, §§4.3.39–4.3.42), and Wimp (1964).

§13.23(v) Other Integrals

Additional integrals involving confluent hypergeometric functions can be found in Apelblat (1983, pp. 388–392), Erdélyi et al. (1954b), Gradshteyn and Ryzhik (2000, §7.6), and Prudnikov et al. (1990, §§1.13, 1.14, 2.19, 4.2.2). See also (13.16.2), (13.16.6), (13.16.7). Generalized orthogonality integrals (33.14.13) and (33.14.15) can be expressed in terms of Whittaker functions via the definitions in that section.