§13.16 Integral Representations

Permalink:: http://dlmf.nist.gov/13.16

§13.16(i) Integrals Along the Real Line
§13.16(ii) Contour Integrals
§13.16(iii) Mellin–Barnes Integrals

§13.16(i) Integrals Along the Real Line

Notes:: Combine §13.4(i) with (13.14.4) and (13.14.5).
Keywords:: Whittaker functions, hypergeometric function, integrals, integrals of modified Bessel functions
Permalink:: http://dlmf.nist.gov/13.16.i

In this subsection see §§10.2(ii), 10.25(ii) for the functions $\mathop{J_{{2\mu}}\/}\nolimits$ , $\mathop{I_{{2\mu}}\/}\nolimits$ , and $\mathop{K_{{2\mu}}\/}\nolimits$ , and §§15.1, 15.2(i) for $\mathop{{{}_{{2}}{\mathbf{F}}_{{1}}}\/}\nolimits$ .

13.16.1 $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(1+2\mu\right)z^{{\mu+\frac{1}{2}}}2^{{-2\mu}}}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu-\kappa\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu+\kappa\right)}\*\int _{{-1}}^{{1}}e^{{\frac{1}{2}zt}}(1+t)^{{\mu-\frac{1}{2}-\kappa}}(1-t)^{{\mu-\frac{1}{2}+\kappa}}dt,$ $\realpart{\mu}+\tfrac{1}{2}>\left|\realpart{\kappa}\right|$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and $z$ : complex variable
Referenced by:: §13.29(iii)
Permalink:: http://dlmf.nist.gov/13.16.E1
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13.16.2 $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(1+2\mu\right)z^{{\lambda}}}{\mathop{\Gamma\/}\nolimits\!\left(1+2\mu-2\lambda\right)\mathop{\Gamma\/}\nolimits\!\left(2\lambda\right)}\*\int _{{0}}^{{1}}\mathop{M_{{\kappa-\lambda,\mu-\lambda}}\/}\nolimits\!\left(zt\right)e^{{\frac{1}{2}z(t-1)}}t^{{\mu-\lambda-\frac{1}{2}}}{(1-t)^{{2\lambda-1}}}dt,$ $\realpart{\mu}+\tfrac{1}{2}>\realpart{\lambda}>0$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and $z$ : complex variable
Referenced by:: §13.23(v)
Permalink:: http://dlmf.nist.gov/13.16.E2
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13.16.3 $\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(1+2\mu\right)}\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)=\frac{\sqrt{z}e^{{\frac{1}{2}z}}}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu+\kappa\right)}\int _{{0}}^{{\infty}}e^{{-t}}t^{{\kappa-\frac{1}{2}}}\mathop{J_{{2\mu}}\/}\nolimits\!\left(2\sqrt{zt}\right)dt,$ $\realpart{(\kappa+\mu)+\tfrac{1}{2}}>0$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker 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/13.16.E3
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13.16.4 $\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(1+2\mu\right)}\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)=\frac{\sqrt{z}e^{{-\frac{1}{2}z}}}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu-\kappa\right)}\*\int _{{0}}^{{\infty}}e^{{-t}}t^{{-\kappa-\frac{1}{2}}}\mathop{I_{{2\mu}}\/}\nolimits\!\left(2\sqrt{zt}\right)dt,$ $\realpart{(\kappa-\mu)-\tfrac{1}{2}}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker 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 and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.16.E4
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13.16.5 $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)=\frac{z^{{\mu+\frac{1}{2}}}2^{{-2\mu}}}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu-\kappa\right)}\*\int _{{1}}^{{\infty}}e^{{-\frac{1}{2}zt}}(t-1)^{{\mu-\frac{1}{2}-\kappa}}(t+1)^{{\mu-\frac{1}{2}+\kappa}}dt,$ $\realpart{\mu}+\tfrac{1}{2}>\realpart{\kappa}$ , $|\mathop{\mathrm{ph}\/}\nolimits{z}|<\frac{1}{2}\pi$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\realpart{}$ : real part and $z$ : complex variable
Referenced by:: §13.29(iii)
Permalink:: http://dlmf.nist.gov/13.16.E5
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13.16.6 $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)=\frac{e^{{-\frac{1}{2}z}}z^{{\kappa+1}}}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu-\kappa\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\mu-\kappa\right)}\*\int _{{0}}^{{\infty}}\frac{\mathop{W_{{-\kappa,\mu}}\/}\nolimits\!\left(t\right)e^{{-\frac{1}{2}t}}t^{{-\kappa-1}}}{t+z}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits{z}|<\pi$ , $\realpart{(\frac{1}{2}+\mu-\kappa)}>\max\left(2\realpart{\mu},0\right)$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $(a,b)$ : open interval, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\realpart{}$ : real part and $z$ : complex variable
Referenced by:: §13.23(v)
Permalink:: http://dlmf.nist.gov/13.16.E6
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13.16.7 $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)=\frac{(-1)^{n}e^{{-\frac{1}{2}z}}z^{{\frac{1}{2}-\mu-n}}}{\mathop{\Gamma\/}\nolimits\!\left(1+2\mu\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\mu-\kappa\right)}\*\int _{0}^{\infty}\frac{\mathop{M_{{-\kappa,\mu}}\/}\nolimits\!\left(t\right)e^{{-\frac{1}{2}t}}t^{{n+\mu-\frac{1}{2}}}}{t+z}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ , $n=0,1,2,\dots$ , $-\realpart{(1+2\mu)}<n<\left|\realpart{\mu}\right|+\realpart{\kappa}<\tfrac{1}{2}$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\realpart{}$ : real part, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §13.23(v)
Permalink:: http://dlmf.nist.gov/13.16.E7
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13.16.8 $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)=\frac{2\sqrt{z}e^{{-\frac{1}{2}z}}}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu-\kappa\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\mu-\kappa\right)}\*\int _{{0}}^{{\infty}}e^{{-t}}t^{{-\kappa-\frac{1}{2}}}\mathop{K_{{2\mu}}\/}\nolimits\!\left(2\sqrt{zt}\right)dt,$ $\realpart{(\mu-\kappa)+\tfrac{1}{2}}>0$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker 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 and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.16.E8
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13.16.9 $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)=e^{{-\frac{1}{2}z}}z^{{\kappa+c}}\*\int _{{0}}^{{\infty}}e^{{-zt}}t^{{c-1}}\mathop{{{}_{{2}}{\mathbf{F}}_{{1}}}\/}\nolimits\!\left({\tfrac{1}{2}+\mu-\kappa,\tfrac{1}{2}-\mu-\kappa\atop c};-t\right)dt,$ $|\mathop{\mathrm{ph}\/}\nolimits{z}|<\frac{1}{2}\pi$ ,

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker 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, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.16.E9
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where $c$ is arbitrary, $\realpart{c}>0$ .

§13.16(ii) Contour Integrals

Keywords:: Whittaker functions
Permalink:: http://dlmf.nist.gov/13.16.ii

For contour integral representations combine (13.14.2) and (13.14.3) with §13.4(ii). See Buchholz (1969, §2.3), Erdélyi et al. (1953a, §6.11.3), and Slater (1960, Chapter 3). See also §13.16(iii).

§13.16(iii) Mellin–Barnes Integrals

Notes:: See Buchholz (1969, §5.4), Erdélyi et al. (1953a, §6.11), and Slater (1960, Chapter 3).
Keywords:: Whittaker functions
Referenced by:: §13.16(ii)
Permalink:: http://dlmf.nist.gov/13.16.iii

If $\tfrac{1}{2}+\mu-\kappa\neq 0,-1,-2,\dots$ , then

13.16.10 $\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(1+2\mu\right)}\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(e^{{\pm\pi i}}z\right)=\frac{e^{{\frac{1}{2}z\pm(\frac{1}{2}+\mu)\pi i}}}{2\pi i\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu-\kappa\right)}\*\int _{{-i\infty}}^{{i\infty}}\frac{\mathop{\Gamma\/}\nolimits\!\left(t-\kappa\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu-t\right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu+t\right)}z^{{t}}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits{z}|<\tfrac{1}{2}\pi$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.16.E10
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where the contour of integration separates the poles of $\mathop{\Gamma\/}\nolimits\!\left(t-\kappa\right)$ from those of $\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu-t\right)$ .

If $\tfrac{1}{2}\pm\mu-\kappa\neq 0,-1,-2,\dots$ , then

13.16.11 $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)=\frac{e^{{-\frac{1}{2}z}}}{2\pi i}\*\int _{{-i\infty}}^{{i\infty}}\frac{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu+t\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\mu+t\right)\mathop{\Gamma\/}\nolimits\!\left(-\kappa-t\right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu-\kappa\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\mu-\kappa\right)}z^{{-t}}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits{z}|<\tfrac{3}{2}\pi$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.16.E11
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where the contour of integration separates the poles of $\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu+t\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\mu+t\right)$ from those of $\mathop{\Gamma\/}\nolimits\!\left(-\kappa-t\right)$ .

13.16.12 $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)=\frac{e^{{\frac{1}{2}z}}}{2\pi i}\int _{{-i\infty}}^{{i\infty}}\frac{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu+t\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\mu+t\right)}{\mathop{\Gamma\/}\nolimits\!\left(1-\kappa+t\right)}z^{{-t}}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits{z}|<\tfrac{1}{2}\pi$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.16.E12
Encodings:: TeX, pMML, png

where the contour of integration passes all the poles of $\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu+t\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\mu+t\right)$ on the right-hand side.

§13.16 Integral Representations

Contents

§13.16(i) Integrals Along the Real Line

§13.16(ii) Contour Integrals

§13.16(iii) Mellin–Barnes Integrals