§13.23 Integrals

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

13.23.1		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{\nu -1}{M}_{\kappa ,\mu }\left(t\right)dt=\frac{\mathrm{\Gamma }\left(\mu +\nu +\frac{1}{2}\right)}{{\left(z+\frac{1}{2}\right)}^{\mu +\nu +\frac{1}{2}}}{}_{2}F_1(\genfrac{}{}{0pt}{}{\frac{1}{2}+\mu -\kappa ,\frac{1}{2}+\mu +\nu }{1+2\mu };\frac{1}{z+\frac{1}{2}}),$$
$\mathrm{\Re }\mu +\nu +\frac{1}{2}>0$, $\mathrm{\Re }z>\frac{1}{2}$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_{2}F_1(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable Keywords: Laplace transform, Mellin transform Referenced by: §13.23(i) Permalink: http://dlmf.nist.gov/13.23.E1 Encodings: TeX, pMML, png See also: Annotations for §13.23(i), §13.23 and Ch.13

13.23.2		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{\mu -\frac{1}{2}}{M}_{\kappa ,\mu }\left(t\right)dt=\mathrm{\Gamma }\left(2\mu +1\right){\left(z+\frac{1}{2}\right)}^{-\kappa -\mu -\frac{1}{2}}{\left(z-\frac{1}{2}\right)}^{\kappa -\mu -\frac{1}{2}},$$
$\mathrm{\Re }\mu >-\frac{1}{2}$, $\mathrm{\Re }z>\frac{1}{2}$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable Keywords: Laplace transform, Mellin transform Permalink: http://dlmf.nist.gov/13.23.E2 Encodings: TeX, pMML, png See also: Annotations for §13.23(i), §13.23 and Ch.13

13.23.3		$$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}t}{t}^{\nu -1}{M}_{\kappa ,\mu }\left(t\right)dt=\frac{\mathrm{\Gamma }\left(\mu +\nu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\kappa -\nu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\nu \right)},$$
.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part Keywords: Laplace transform, Mellin transform Referenced by: §13.23(i) Permalink: http://dlmf.nist.gov/13.23.E3 Encodings: TeX, pMML, png See also: Annotations for §13.23(i), §13.23 and Ch.13

13.23.4		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{\nu -1}{W}_{\kappa ,\mu }\left(t\right)dt=\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\nu \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu +\nu \right){}_2\mathbf{F}_1(\genfrac{}{}{0pt}{}{\frac{1}{2}-\mu +\nu ,\frac{1}{2}+\mu +\nu }{\nu -\kappa +1};\frac{1}{2}-z),$$
$\mathrm{\Re }\left(\nu +\frac{1}{2}\right)>|\mathrm{\Re }\mu |$, $\mathrm{\Re }z>-\frac{1}{2}$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, ${}_p\mathbf{F}_q(\mathbf{a};\mathbf{b};z)$ or ${}_p\mathbf{F}_q(\genfrac{}{}{0pt}{}{\mathbf{a}}{\mathbf{b}};z)$: scaled (or Olver’s) generalized hypergeometric function, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable Keywords: Laplace transform, Mellin transform Referenced by: §13.23(i) Permalink: http://dlmf.nist.gov/13.23.E4 Encodings: TeX, pMML, png See also: Annotations for §13.23(i), §13.23 and Ch.13

13.23.5		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{\frac{1}{2}t}{t}^{\nu -1}{W}_{\kappa ,\mu }\left(t\right)dt=\frac{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\nu \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu +\nu \right)\mathrm{\Gamma }\left(-\kappa -\nu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\kappa \right)},$$
.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part Keywords: Mellin transform Referenced by: §13.23(i) Permalink: http://dlmf.nist.gov/13.23.E5 Encodings: TeX, pMML, png See also: Annotations for §13.23(i), §13.23 and Ch.13

13.23.6		$$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{zt+\frac{1}{2}{t}^{-1}}{t}^{\kappa }{M}_{\kappa ,\mu }\left(t^{-1}\right)dt=\frac{z^{-\kappa -\frac{1}{2}}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{I}_{2\mu }\left(2\sqrt{z}\right),$$
$\mathrm{\Re }z>0$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part and $z$: complex variable Keywords: Mellin transform Permalink: http://dlmf.nist.gov/13.23.E6 Encodings: TeX, pMML, png See also: Annotations for §13.23(i), §13.23 and Ch.13

13.23.7		$$\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{zt+\frac{1}{2}{t}^{-1}}{t}^{\kappa }{W}_{\kappa ,\mu }\left(t^{-1}\right)dt=\frac{2z^{-\kappa -\frac{1}{2}}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\kappa \right)}{K}_{2\mu }\left(2\sqrt{z}\right),$$
$\mathrm{\Re }z>0$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part and $z$: complex variable Keywords: Mellin transform Permalink: http://dlmf.nist.gov/13.23.E7 Encodings: TeX, pMML, png See also: Annotations for §13.23(i), §13.23 and Ch.13

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).

13.23.8		$$\begin{array}{l}\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{\int }_0^{\mathrm{\infty }}\cos \left(2xt\right){\mathrm{e}}^{-\frac{1}{2}{t}^2}{t}^{-2\mu -1}{M}_{\kappa ,\mu }\left(t^2\right)dt\\ =\frac{\sqrt{\pi }{\mathrm{e}}^{-\frac{1}{2}{x}^2}{x}^{\mu +\kappa -1}}{2\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)}{W}_{\frac{1}{2}\kappa -\frac{3}{2}\mu ,\frac{1}{2}\kappa +\frac{1}{2}\mu }\left(x^2\right),\end{array}$$
$\mathrm{\Re }\left(\kappa +\mu \right)>-\frac{1}{2}$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable Permalink: http://dlmf.nist.gov/13.23.E8 Encodings: TeX, pMML, png See also: Annotations for §13.23(ii), §13.23 and Ch.13

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

For the notation see §10.2(ii).

13.23.9		$$\begin{array}{l}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}t}{t}^{\mu -\frac{1}{2}(\nu +1)}{M}_{\kappa ,\mu }\left(t\right)J_{\nu }\left(2\sqrt{xt}\right)dt\\ =\frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}-\mu +\kappa +\nu \right)}{\mathrm{e}}^{-\frac{1}{2}x}{x}^{\frac{1}{2}(\kappa -\mu -\frac{3}{2})}{M}_{\frac{1}{2}(\kappa +3\mu -\nu +\frac{1}{2}),\frac{1}{2}(\kappa -\mu +\nu -\frac{1}{2})}\left(x\right),\end{array}$$
$x>0$, ,
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable Permalink: http://dlmf.nist.gov/13.23.E9 Encodings: TeX, pMML, png See also: Annotations for §13.23(iii), §13.23 and Ch.13

13.23.10		$$\begin{array}{l}\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}t}{t}^{\frac{1}{2}(\nu -1)-\mu }{M}_{\kappa ,\mu }\left(t\right)J_{\nu }\left(2\sqrt{xt}\right)dt\\ =\frac{{\mathrm{e}}^{-\frac{1}{2}x}{x}^{\frac{1}{2}(\kappa +\mu -\frac{3}{2})}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)}{W}_{\frac{1}{2}(\kappa -3\mu +\nu +\frac{1}{2}),\frac{1}{2}(\kappa +\mu -\nu -\frac{1}{2})}\left(x\right),\end{array}$$
$x>0$, .
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable Permalink: http://dlmf.nist.gov/13.23.E10 Encodings: TeX, pMML, png See also: Annotations for §13.23(iii), §13.23 and Ch.13

13.23.11		$$\begin{array}{l}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{\frac{1}{2}t}{t}^{\frac{1}{2}(\nu -1)-\mu }{W}_{\kappa ,\mu }\left(t\right)J_{\nu }\left(2\sqrt{xt}\right)dt\\ =\frac{\mathrm{\Gamma }\left(\nu -2\mu +1\right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{\mathrm{e}}^{\frac{1}{2}x}{x}^{\frac{1}{2}(\mu -\kappa -\frac{3}{2})}{W}_{\frac{1}{2}(\kappa +3\mu -\nu -\frac{1}{2}),\frac{1}{2}(\kappa -\mu +\nu +\frac{1}{2})}\left(x\right),\end{array}$$
$x>0$, ,
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable Permalink: http://dlmf.nist.gov/13.23.E11 Encodings: TeX, pMML, png See also: Annotations for §13.23(iii), §13.23 and Ch.13

13.23.12		$$\begin{array}{l}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}t}{t}^{\frac{1}{2}(\nu -1)-\mu }{W}_{\kappa ,\mu }\left(t\right)J_{\nu }\left(2\sqrt{xt}\right)dt\\ =\frac{\mathrm{\Gamma }\left(\nu -2\mu +1\right)}{\mathrm{\Gamma }\left(\frac{3}{2}-\mu -\kappa +\nu \right)}{\mathrm{e}}^{-\frac{1}{2}x}{x}^{\frac{1}{2}(\mu +\kappa -\frac{3}{2})}{M}_{\frac{1}{2}(\kappa -3\mu +\nu +\frac{1}{2}),\frac{1}{2}(\nu -\mu -\kappa +\frac{1}{2})}\left(x\right),\end{array}$$
$x>0$, .
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable Permalink: http://dlmf.nist.gov/13.23.E12 Encodings: TeX, pMML, png See also: Annotations for §13.23(iii), §13.23 and Ch.13

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).

Let $f(x)$ be absolutely integrable on the interval $[r,R]$ for all positive , $f(x)=O\left(x^{{\rho }_0}\right)$ as $x\to 0+$, and $f(x)=O\left({\mathrm{e}}^{-{\rho }_{1}x}\right)$ as $x\to +\mathrm{\infty }$, where ${\rho }_1>\frac{1}{2}$. Then for $\mu$ in the half-plane $\mathrm{\Re }\mu \ge {\mu }_1>\max (-{\rho }_0,\mathrm{\Re }\kappa -\frac{1}{2})$

13.23.13		$g(\mu )$	$={\displaystyle \frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}}{\displaystyle {\int }_0^{\mathrm{\infty }}}f(x)x^{-\frac{3}{2}}{M}_{\kappa ,\mu }\left(x\right)dx,$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$, $\int$: integral and $x$: real variable Permalink: http://dlmf.nist.gov/13.23.E13 Encodings: TeX, pMML, png See also: Annotations for §13.23(iv), §13.23 and Ch.13
13.23.14		$f(x)$	$={\displaystyle \frac{1}{\pi \mathrm{i}\sqrt{x}}}{\displaystyle {\int }_{{\mu }_1-\mathrm{i}\mathrm{\infty }}^{{\mu }_1+\mathrm{i}\mathrm{\infty }}}\mu g(\mu )\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)W_{\kappa ,\mu }\left(x\right)d\mu .$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{i}$: imaginary unit, $\int$: integral and $x$: real variable Permalink: http://dlmf.nist.gov/13.23.E14 Encodings: TeX, pMML, png See also: Annotations for §13.23(iv), §13.23 and Ch.13

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

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.