13 Confluent Hypergeometric Functions Kummer Functions 13.9 Zeros 13.11 Series

§13.10 Integrals

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Referenced by:: §12.12
Permalink:: http://dlmf.nist.gov/13.10
See also:: Annotations for Ch.13

§13.10(i) Indefinite Integrals

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Keywords:: Kummer functions, indefinite, integrals
Permalink:: http://dlmf.nist.gov/13.10.i
See also:: Annotations for §13.10 and Ch.13

When $a\neq 1$ ,

13.10.1		$\int{\mathbf{M}}\left(a,b,z\right)\,\mathrm{d}z=\frac{1}{a-1}{\mathbf{M}}\left% (a-1,b-1,z\right),$
ⓘ Symbols: ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : 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		$\int U\left(a,b,z\right)\,\mathrm{d}z=-\frac{1}{a-1}U\left(a-1,b-1,z\right).$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : 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).

§13.10(ii) Laplace Transforms

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Keywords:: Kummer functions, Laplace transforms, integrals
Notes:: See Erdélyi et al. (1953a, §6.10) and Slater (1960, Chapter 3).
Permalink:: http://dlmf.nist.gov/13.10.ii
Clarification (effective with 1.2.1):: Just below (13.10.9), we added “For the particular loop contour, see Figure 5.9.1.”
See also:: Annotations for §13.10 and Ch.13

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

13.10.3		$\int_{0}^{\infty}e^{-zt}t^{b-1}{\mathbf{M}}\left(a,c,kt\right)\,\mathrm{d}t=% \Gamma\left(b\right)z^{-b}{{}_{2}{\mathbf{F}}_{1}}\left(a,b;c;\ifrac{k}{z}% \right),$
$\Re b>0$ , $\Re z>\max\left(\Re k,0\right)$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ or ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{% \mathbf{b}}};\NVar{z}\right)$ : scaled (or Olver’s) generalized hypergeometric function, $\int$ : integral, $\Re$ : 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		$\int_{0}^{\infty}e^{-zt}t^{b-1}{\mathbf{M}}\left(a,b,t\right)\,\mathrm{d}t=z^{% -b}\left(1-\frac{1}{z}\right)^{-a},$
$\Re b>0$ , $\Re z>1$ ,
ⓘ Symbols: ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : 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		$\int_{0}^{\infty}e^{-t}t^{b-1}{\mathbf{M}}\left(a,c,t\right)\,\mathrm{d}t=% \frac{\Gamma\left(b\right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a\right)% \Gamma\left(c-b\right)},$
$\Re\left(c-a\right)>\Re b>0$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $\Re$ : 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		$\int_{0}^{\infty}e^{-zt-t^{2}}t^{2b-2}{\mathbf{M}}\left(a,b,t^{2}\right)\,% \mathrm{d}t=\tfrac{1}{2}\pi^{-\frac{1}{2}}\Gamma\left(b-\tfrac{1}{2}\right)U% \left(b-\tfrac{1}{2},a+\tfrac{1}{2},\tfrac{1}{4}z^{2}\right),$
$\Re b>\tfrac{1}{2}$ , $\Re z>0$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : 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		$\int_{0}^{\infty}e^{-zt}t^{b-1}U\left(a,c,t\right)\,\mathrm{d}t=\Gamma\left(b% \right)\Gamma\left(b-c+1\right)\*z^{-b}{{}_{2}{\mathbf{F}}_{1}}\left(a,b;a+b-c% +1;1-\frac{1}{z}\right),$
$\Re b>\max\left(\Re c-1,0\right)$ , $\Re z>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ or ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{% \mathbf{b}}};\NVar{z}\right)$ : scaled (or Olver’s) generalized hypergeometric function, $\int$ : integral, $\Re$ : 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

Loop Integrals

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See also:: Annotations for §13.10(ii), §13.10 and Ch.13

13.10.8		$\frac{1}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{tz}t^{-a}{\mathbf{M}}\left(a,b% ,\ifrac{y}{t}\right)\,\mathrm{d}t=\frac{1}{\Gamma\left(a\right)}z^{\frac{1}{2}% (2a-b-1)}y^{\frac{1}{2}(1-b)}I_{b-1}\left(2\sqrt{zy}\right),$
$\Re z>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\Re$ : real part, $y$ : real variable and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.10.E8 Encodings: TeX, pMML, png See also: Annotations for §13.10(ii), §13.10(ii), §13.10 and Ch.13

13.10.9		$\frac{1}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{tz}t^{-a}U\left(a,b,\ifrac{y}{% t}\right)\,\mathrm{d}t=\frac{2z^{\frac{1}{2}(2a-b-1)}y^{\frac{1}{2}(1-b)}}{% \Gamma\left(a\right)\Gamma\left(a-b+1\right)}K_{b-1}\left(2\sqrt{zy}\right),$
$\Re z>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\Re$ : real part, $y$ : real variable and $z$ : complex variable Referenced by: §13.10(ii), Erratum (V1.2.1) for Figure 5.9.1 Permalink: http://dlmf.nist.gov/13.10.E9 Encodings: TeX, pMML, png See also: Annotations for §13.10(ii), §13.10(ii), §13.10 and Ch.13

For the particular loop contour, see Figure 5.9.1.

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

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Keywords:: Kummer functions, Mellin transforms, integrals
Notes:: See Slater (1960, Chapter 3).
Permalink:: http://dlmf.nist.gov/13.10.iii
See also:: Annotations for §13.10 and Ch.13

13.10.10		$\int_{0}^{\infty}t^{\lambda-1}{\mathbf{M}}\left(a,b,-t\right)\,\mathrm{d}t=% \frac{\Gamma\left(\lambda\right)\Gamma\left(a-\lambda\right)}{\Gamma\left(a% \right)\Gamma\left(b-\lambda\right)},$
$0<\Re\lambda<\Re a$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\Re$ : 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		$\int_{0}^{\infty}t^{\lambda-1}U\left(a,b,t\right)\,\mathrm{d}t=\frac{\Gamma% \left(\lambda\right)\Gamma\left(a-\lambda\right)\Gamma\left(\lambda-b+1\right)% }{\Gamma\left(a\right)\Gamma\left(a-b+1\right)},$
$\max\left(\Re b-1,0\right)<\Re\lambda<\Re a$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\Re$ : 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(iv) Fourier Transforms

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Keywords:: Fourier transforms, Kummer functions, integrals
Notes:: See Erdélyi et al. (1953a, §6.15.2).
Permalink:: http://dlmf.nist.gov/13.10.iv
See also:: Annotations for §13.10 and Ch.13

13.10.12		$\int_{0}^{\infty}\cos\left(2xt\right){\mathbf{M}}\left(a,b,-t^{2}\right)\,% \mathrm{d}t=\frac{\sqrt{\pi}}{2\Gamma\left(a\right)}x^{2a-1}e^{-x^{2}}U\left(b% -\tfrac{1}{2},a+\tfrac{1}{2},x^{2}\right),$
$\Re a>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : 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).

§13.10(v) Hankel Transforms

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Keywords:: Hankel transforms, Kummer functions, integrals
Notes:: See Buchholz (1969, §11.1), including the references given there. (13.10.14) and (13.10.16) are from Erdélyi et al. (1954b, §8.18). For (13.10.14) substitute the integral (13.4.1) for ${\mathbf{M}}\left(a,b,t\right)$ , interchange the order of integration, then apply (13.4.3) and (13.6.1) followed by (13.4.4). For (13.10.16) interchange the roles of (13.4.1) and (13.4.4).
Referenced by:: §13.23(iii)
Permalink:: http://dlmf.nist.gov/13.10.v
See also:: Annotations for §13.10 and Ch.13

For the notation see §10.2(ii).

13.10.13		$\int_{0}^{\infty}e^{-t}t^{b-1-\frac{1}{2}\nu}{\mathbf{M}}\left(a,b,t\right)J_{% \nu}\left(2\sqrt{xt}\right)\,\mathrm{d}t=x^{-a+\frac{1}{2}\nu}e^{-x}{\mathbf{M% }}\left(\nu-b+1,\nu-a+1,x\right),$
$x>0$ , $2\Re a<\Re\nu+\tfrac{5}{2}$ , $\Re b>0$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : 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		$\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}{\mathbf{M}}\left(a,b,t\right)J_{\nu}% \left(2\sqrt{xt}\right)\,\mathrm{d}t=\frac{x^{\frac{1}{2}\nu}e^{-x}}{\Gamma% \left(b-a\right)}U\left(a,a-b+\nu+2,x\right),$
$x>0$ , $-1<\Re\nu<2\Re\left(b-a\right)-\tfrac{1}{2}$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : 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		$\int_{0}^{\infty}t^{\frac{1}{2}\nu}U\left(a,b,t\right)J_{\nu}\left(2\sqrt{xt}% \right)\,\mathrm{d}t=\frac{\Gamma\left(\nu-b+2\right)}{\Gamma\left(a\right)}x^% {\frac{1}{2}\nu}U\left(\nu-b+2,\nu-a+2,x\right),$
$x>0$ , $\max\left(\Re b-2,-1\right)<\Re\nu<2\Re a+\tfrac{1}{2}$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : 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		$\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}U\left(a,b,t\right)J_{\nu}\left(2% \sqrt{xt}\right)\,\mathrm{d}t=\Gamma\left(\nu-b+2\right)x^{\frac{1}{2}\nu}e^{-% x}{\mathbf{M}}\left(a,a-b+\nu+2,x\right),$
$x>0$ , $\max\left(\Re b-2,-1\right)<\Re\nu$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : 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).

§13.10(vi) Other Integrals

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Keywords:: Kummer functions, compendia, integrals
Referenced by:: Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)
Permalink:: http://dlmf.nist.gov/13.10.vi
Addition (effective with 1.0.14):: A sentence was added at the end of this section to provide cross-references to certain generalized orthogonality integrals in terms of Kummer functions.
See also:: Annotations for §13.10 and Ch.13

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 (2015, §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.

13.9 Zeros 13.11 Series

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