§13.10 Integrals

Referenced by:: ¶ ‣ §12.12
Permalink:: http://dlmf.nist.gov/13.10

§13.10(i) Indefinite Integrals
§13.10(ii) Laplace Transforms
§13.10(iii) Mellin Transforms
§13.10(iv) Fourier Transforms
§13.10(v) Hankel Transforms
§13.10(vi) Other Integrals

§13.10(i) Indefinite Integrals

Keywords:: Kummer functions, integrals
Permalink:: http://dlmf.nist.gov/13.10.i

When $a\neq 1$ ,

13.10.1 $\int\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)dz=\frac{1}{a-1}\mathop{{\mathbf{M}}\/}\nolimits\!\left(a-1,b-1,z\right),$

Symbols:: $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $dx$ : differential of $x$ , $\int$ : integral and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.10.E1
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13.10.2 $\int\mathop{U\/}\nolimits\!\left(a,b,z\right)dz=-\frac{1}{a-1}\mathop{U\/}\nolimits\!\left(a-1,b-1,z\right).$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $dx$ : differential of $x$ , $\int$ : integral and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.10.E2
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Other formulas of this kind can be constructed by inversion of the differentiation formulas given in §13.3(ii).

§13.10(ii) Laplace Transforms

Notes:: See Erdélyi et al. (1953a, §6.10) and Slater (1960, Chapter 3).
Keywords:: Kummer functions, integrals
Permalink:: http://dlmf.nist.gov/13.10.ii

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

13.10.3 $\int _{{0}}^{{\infty}}e^{{-zt}}t^{{b-1}}\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,c,kt\right)dt=\mathop{\Gamma\/}\nolimits\!\left(b\right)z^{{-b}}\mathop{{{}_{{2}}{\mathbf{F}}_{{1}}}\/}\nolimits\!\left(a,b;c;\ifrac{k}{z}\right),$ $\realpart{b}>0$ , $\realpart{z}>\max\left(\realpart{k},0\right)$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{{{}_{{p}}{\mathbf{F}}_{{q}}}\/}\nolimits\!\left({\mathbf{a}\atop\mathbf{b}};z\right)$ : scaled (or Olver’s) generalized hypergeometric function, $\int$ : integral, $(a,b)$ : open interval, $\realpart{}$ : real part and $z$ : complex variable
Referenced by:: §8.14
Permalink:: http://dlmf.nist.gov/13.10.E3
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13.10.4 $\int _{{0}}^{{\infty}}e^{{-zt}}t^{{b-1}}\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,t\right)dt=z^{{-b}}\left(1-\frac{1}{z}\right)^{{-a}},$ $\realpart{b}>0$ , $\realpart{z}>1$ ,

Symbols:: $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and $z$ : complex variable
Referenced by:: §8.14
Permalink:: http://dlmf.nist.gov/13.10.E4
Encodings:: TeX, pMML, png

13.10.5 $\int _{{0}}^{{\infty}}e^{{-t}}t^{{b-1}}\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,c,t\right)dt=\frac{\mathop{\Gamma\/}\nolimits\!\left(b\right)\mathop{\Gamma\/}\nolimits\!\left(c-a-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(c-a\right)\mathop{\Gamma\/}\nolimits\!\left(c-b\right)},$ $\realpart{(c-a)}>\realpart{b}>0$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $\realpart{}$ : real part
Permalink:: http://dlmf.nist.gov/13.10.E5
Encodings:: TeX, pMML, png

13.10.6 $\int _{{0}}^{{\infty}}e^{{-zt-t^{2}}}t^{{2b-2}}\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,t^{2}\right)dt=\tfrac{1}{2}\pi^{{-\frac{1}{2}}}\mathop{\Gamma\/}\nolimits\!\left(b-\tfrac{1}{2}\right)\mathop{U\/}\nolimits\!\left(b-\tfrac{1}{2},a+\tfrac{1}{2},\tfrac{1}{4}z^{2}\right),$ $\realpart{b}>\tfrac{1}{2}$ , $\realpart{z}>0$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.10.E6
Encodings:: TeX, pMML, png

13.10.7 $\int _{{0}}^{{\infty}}e^{{-zt}}t^{{b-1}}\mathop{U\/}\nolimits\!\left(a,c,t\right)dt=\mathop{\Gamma\/}\nolimits\!\left(b\right)\mathop{\Gamma\/}\nolimits\!\left(b-c+1\right)\* z^{{-b}}\mathop{{{}_{{2}}{\mathbf{F}}_{{1}}}\/}\nolimits\!\left(a,b;a+b-c+1;1-\frac{1}{z}\right),$ $\realpart{b}>\max\left(\realpart{c-1},0\right)$ , $\realpart{z}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{{{}_{{p}}{\mathbf{F}}_{{q}}}\/}\nolimits\!\left({\mathbf{a}\atop\mathbf{b}};z\right)$ : scaled (or Olver’s) generalized hypergeometric function, $\int$ : integral, $(a,b)$ : open interval, $\realpart{}$ : real part and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.10.E7
Encodings:: TeX, pMML, png

¶ Loop Integrals

13.10.8 $\frac{1}{2\pi i}\int _{{-\infty}}^{{(0+)}}e^{{tz}}t^{{-a}}\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,\ifrac{y}{t}\right)dt=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(a\right)}z^{{\frac{1}{2}(2a-b-1)}}y^{{\frac{1}{2}(1-b)}}\mathop{I_{{b-1}}\/}\nolimits\!\left(2\sqrt{zy}\right),$ $\realpart{z}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part, $y$ : real variable and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.10.E8
Encodings:: TeX, pMML, png

13.10.9 $\frac{1}{2\pi i}\int _{{-\infty}}^{{(0+)}}e^{{tz}}t^{{-a}}\mathop{U\/}\nolimits\!\left(a,b,\ifrac{y}{t}\right)dt=\frac{2z^{{\frac{1}{2}(2a-b-1)}}y^{{\frac{1}{2}(1-b)}}}{\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)}\mathop{K_{{b-1}}\/}\nolimits\!\left(2\sqrt{zy}\right),$ $\realpart{z}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part, $y$ : real variable and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.10.E9
Encodings:: TeX, pMML, png

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

Notes:: See Slater (1960, Chapter 3).
Keywords:: Kummer functions, integrals
Permalink:: http://dlmf.nist.gov/13.10.iii

13.10.10 $\int _{{0}}^{{\infty}}t^{{\lambda-1}}\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,-t\right)dt=\frac{\mathop{\Gamma\/}\nolimits\!\left(\lambda\right)\mathop{\Gamma\/}\nolimits\!\left(a-\lambda\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(b-\lambda\right)},$ $0<\realpart{\lambda}<\realpart{a}$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $dx$ : differential of $x$ , $\int$ : integral and $\realpart{}$ : real part
Referenced by:: §8.14
Permalink:: http://dlmf.nist.gov/13.10.E10
Encodings:: TeX, pMML, png

13.10.11 $\int _{{0}}^{{\infty}}t^{{\lambda-1}}\mathop{U\/}\nolimits\!\left(a,b,t\right)dt=\frac{\mathop{\Gamma\/}\nolimits\!\left(\lambda\right)\mathop{\Gamma\/}\nolimits\!\left(a-\lambda\right)\mathop{\Gamma\/}\nolimits\!\left(\lambda-b+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)},$ $\max\left(\realpart{b-1},0\right)<\realpart{\lambda}<\realpart{a}$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $dx$ : differential of $x$ , $\int$ : integral, $(a,b)$ : open interval and $\realpart{}$ : real part
Referenced by:: §8.14
Permalink:: http://dlmf.nist.gov/13.10.E11
Encodings:: TeX, pMML, png

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

Notes:: See Erdélyi et al. (1953a, §6.15.2).
Keywords:: Kummer functions, integrals
Permalink:: http://dlmf.nist.gov/13.10.iv

13.10.12 $\int _{{0}}^{{\infty}}\mathop{\cos\/}\nolimits\!\left(2xt\right)\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,-t^{2}\right)dt=\frac{\sqrt{\pi}}{2\mathop{\Gamma\/}\nolimits\!\left(a\right)}x^{{2a-1}}e^{{-x^{2}}}\mathop{U\/}\nolimits\!\left(b-\tfrac{1}{2},a+\tfrac{1}{2},x^{2}\right),$ $\realpart{a}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and $x$ : real variable
Permalink:: http://dlmf.nist.gov/13.10.E12
Encodings:: TeX, pMML, png

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

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 $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,t\right)$ , 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).
Keywords:: Kummer functions, integrals
Referenced by:: §13.23(iii)
Permalink:: http://dlmf.nist.gov/13.10.v

For the notation see §10.2(ii).

13.10.13 $\int _{{0}}^{{\infty}}e^{{-t}}t^{{b-1-\frac{1}{2}\nu}}\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,t\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(2\sqrt{xt}\right)dt=x^{{-a+\frac{1}{2}\nu}}e^{{-x}}\mathop{{\mathbf{M}}\/}\nolimits\!\left(\nu-b+1,\nu-a+1,x\right),$ $x>0$ , $2\realpart{a}<\realpart{\nu}+\tfrac{5}{2}$ , $\realpart{b}>0$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and $x$ : real variable
Permalink:: http://dlmf.nist.gov/13.10.E13
Encodings:: TeX, pMML, png

13.10.14 $\int _{{0}}^{{\infty}}e^{{-t}}t^{{\frac{1}{2}\nu}}\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,t\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(2\sqrt{xt}\right)dt=\frac{x^{{\frac{1}{2}\nu}}e^{{-x}}}{\mathop{\Gamma\/}\nolimits\!\left(b-a\right)}\mathop{U\/}\nolimits\!\left(a,a-b+\nu+2,x\right),$ $x>0$ , $-1<\realpart{\nu}<2\realpart{(b-a)}-\tfrac{1}{2}$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and $x$ : real variable
Referenced by:: §13.10(v)
Permalink:: http://dlmf.nist.gov/13.10.E14
Encodings:: TeX, pMML, png

13.10.15 $\int _{{0}}^{{\infty}}t^{{\frac{1}{2}\nu}}\mathop{U\/}\nolimits\!\left(a,b,t\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(2\sqrt{xt}\right)dt=\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu-b+2\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)}x^{{\frac{1}{2}\nu}}\mathop{U\/}\nolimits\!\left(\nu-b+2,\nu-a+2,x\right),$ $x>0$ , $\max\left(\realpart{b-2},-1\right)<\realpart{\nu}<2\realpart{a}+\tfrac{1}{2}$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $dx$ : differential of $x$ , $\int$ : integral, $(a,b)$ : open interval, $\realpart{}$ : real part and $x$ : real variable
Permalink:: http://dlmf.nist.gov/13.10.E15
Encodings:: TeX, pMML, png

13.10.16 $\int _{{0}}^{{\infty}}e^{{-t}}t^{{\frac{1}{2}\nu}}\mathop{U\/}\nolimits\!\left(a,b,t\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(2\sqrt{xt}\right)dt=\mathop{\Gamma\/}\nolimits\!\left(\nu-b+2\right)x^{{\frac{1}{2}\nu}}e^{{-x}}\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,a-b+\nu+2,x\right),$ $x>0$ , $\max\left(\realpart{b-2},-1\right)<\realpart{\nu}$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $(a,b)$ : open interval, $\realpart{}$ : real part and $x$ : real variable
Referenced by:: §13.10(v)
Permalink:: http://dlmf.nist.gov/13.10.E16
Encodings:: TeX, pMML, png

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

Keywords:: Kummer functions, integrals
Permalink:: http://dlmf.nist.gov/13.10.vi

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).