§13.10 Integrals

When $a\ne 1$,

13.10.1		$$\int \mathbf{M}(a,b,z)dz=\frac{1}{a-1}\mathbf{M}(a-1,b-1,z),$$
ⓘ Symbols: $\mathbf{M}(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$, $\int$: integral and $z$: complex variable Permalink: http://dlmf.nist.gov/13.10.E1 Encodings: TeX, pMML, png See also: Annotations for §13.10(i), §13.10 and Ch.13

13.10.2		$$\int U(a,b,z)dz=-\frac{1}{a-1}U(a-1,b-1,z).$$
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, $dx$: differential of $x$, $\int$: integral and $z$: complex variable Permalink: http://dlmf.nist.gov/13.10.E2 Encodings: TeX, pMML, png See also: Annotations for §13.10(i), §13.10 and Ch.13

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

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

13.10.3		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{b-1}\mathbf{M}(a,c,kt)dt=\mathrm{\Gamma }\left(b\right)z^{-b}{}_2\mathbf{F}_1(a,b;c;k/z),$$
$\mathrm{\Re }b>0$, $\mathrm{\Re }z>\max (\mathrm{\Re }k,0)$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathbf{M}(a,b,z)$: Olver’s 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 Referenced by: §8.14 Permalink: http://dlmf.nist.gov/13.10.E3 Encodings: TeX, pMML, png See also: Annotations for §13.10(ii), §13.10 and Ch.13

13.10.4		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{b-1}\mathbf{M}(a,b,t)dt=z^{-b}{\left(1-\frac{1}{z}\right)}^{-a},$$
$\mathrm{\Re }b>0$, $\mathrm{\Re }z>1$,
ⓘ Symbols: $\mathbf{M}(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable Referenced by: §8.14 Permalink: http://dlmf.nist.gov/13.10.E4 Encodings: TeX, pMML, png See also: Annotations for §13.10(ii), §13.10 and Ch.13

13.10.5		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{b-1}\mathbf{M}(a,c,t)dt=\frac{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-a-b\right)}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)},$$
$\mathrm{\Re }\left(c-a\right)>\mathrm{\Re }b>0$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathbf{M}(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part Permalink: http://dlmf.nist.gov/13.10.E5 Encodings: TeX, pMML, png See also: Annotations for §13.10(ii), §13.10 and Ch.13

13.10.6		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt-t^2}{t}^{2b-2}\mathbf{M}(a,b,t^2)dt=\frac{1}{2}{\pi }^{-\frac{1}{2}}\mathrm{\Gamma }\left(b-\frac{1}{2}\right)U(b-\frac{1}{2},a+\frac{1}{2},\frac{1}{4}{z}^2),$$
$\mathrm{\Re }b>\frac{1}{2}$, $\mathrm{\Re }z>0$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathbf{M}(a,b,z)$: Olver’s 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, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable Permalink: http://dlmf.nist.gov/13.10.E6 Encodings: TeX, pMML, png See also: Annotations for §13.10(ii), §13.10 and Ch.13

13.10.7		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{b-1}U(a,c,t)dt=\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(b-c+1\right)z^{-b}{}_2\mathbf{F}_1(a,b;a+b-c+1;1-\frac{1}{z}),$$
$\mathrm{\Re }b>\max (\mathrm{\Re }c-1,0)$, $\mathrm{\Re }z>0$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer 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 Permalink: http://dlmf.nist.gov/13.10.E7 Encodings: TeX, pMML, png See also: Annotations for §13.10(ii), §13.10 and Ch.13

13.10.10		$${\int }_0^{\mathrm{\infty }}{t}^{\lambda -1}\mathbf{M}(a,b,-t)dt=\frac{\mathrm{\Gamma }\left(\lambda \right)\mathrm{\Gamma }\left(a-\lambda \right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b-\lambda \right)},$$
,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathbf{M}(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$, $\int$: integral and $\mathrm{\Re }$: real part Keywords: Mellin transform Referenced by: §8.14 Permalink: http://dlmf.nist.gov/13.10.E10 Encodings: TeX, pMML, png See also: Annotations for §13.10(iii), §13.10 and Ch.13

13.10.11		$${\int }_0^{\mathrm{\infty }}{t}^{\lambda -1}U(a,b,t)dt=\frac{\mathrm{\Gamma }\left(\lambda \right)\mathrm{\Gamma }\left(a-\lambda \right)\mathrm{\Gamma }\left(\lambda -b+1\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(a-b+1\right)},$$
.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $dx$: differential of $x$, $\int$: integral and $\mathrm{\Re }$: real part Keywords: Mellin transform Referenced by: §8.14 Permalink: http://dlmf.nist.gov/13.10.E11 Encodings: TeX, pMML, png See also: Annotations for §13.10(iii), §13.10 and Ch.13

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).

13.10.12		$${\int }_0^{\mathrm{\infty }}\cos \left(2xt\right)\mathbf{M}(a,b,-t^2)dt=\frac{\sqrt{\pi }}{2\mathrm{\Gamma }\left(a\right)}{x}^{2a-1}{\mathrm{e}}^{-x^2}U(b-\frac{1}{2},a+\frac{1}{2},x^2),$$
$\mathrm{\Re }a>0$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathbf{M}(a,b,z)$: Olver’s 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/13.10.E12 Encodings: TeX, pMML, png See also: Annotations for §13.10(iv), §13.10 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).

13.10.13		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{b-1-\frac{1}{2}\nu }\mathbf{M}(a,b,t)J_{\nu }\left(2\sqrt{xt}\right)dt=x^{-a+\frac{1}{2}\nu }{\mathrm{e}}^{-x}\mathbf{M}(\nu -b+1,\nu -a+1,x),$$
$x>0$, , $\mathrm{\Re }b>0$,
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathbf{M}(a,b,z)$: Olver’s 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/13.10.E13 Encodings: TeX, pMML, png See also: Annotations for §13.10(v), §13.10 and Ch.13

13.10.14		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{\frac{1}{2}\nu }\mathbf{M}(a,b,t)J_{\nu }\left(2\sqrt{xt}\right)dt=\frac{x^{\frac{1}{2}\nu }{\mathrm{e}}^{-x}}{\mathrm{\Gamma }\left(b-a\right)}U(a,a-b+\nu +2,x),$$
$x>0$, ,
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathbf{M}(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable Referenced by: §13.10(v) Permalink: http://dlmf.nist.gov/13.10.E14 Encodings: TeX, pMML, png See also: Annotations for §13.10(v), §13.10 and Ch.13

13.10.15		$${\int }_0^{\mathrm{\infty }}{t}^{\frac{1}{2}\nu }U(a,b,t)J_{\nu }\left(2\sqrt{xt}\right)dt=\frac{\mathrm{\Gamma }\left(\nu -b+2\right)}{\mathrm{\Gamma }\left(a\right)}{x}^{\frac{1}{2}\nu }U(\nu -b+2,\nu -a+2,x),$$
$x>0$, ,
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $dx$: differential of $x$, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable Permalink: http://dlmf.nist.gov/13.10.E15 Encodings: TeX, pMML, png See also: Annotations for §13.10(v), §13.10 and Ch.13

13.10.16		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{\frac{1}{2}\nu }U(a,b,t)J_{\nu }\left(2\sqrt{xt}\right)dt=\mathrm{\Gamma }\left(\nu -b+2\right)x^{\frac{1}{2}\nu }{\mathrm{e}}^{-x}\mathbf{M}(a,a-b+\nu +2,x),$$
$x>0$, .
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathbf{M}(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable Referenced by: §13.10(v) Permalink: http://dlmf.nist.gov/13.10.E16 Encodings: TeX, pMML, png See also: Annotations for §13.10(v), §13.10 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).

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.