13 Confluent Hypergeometric Functions Whittaker Functions 13.15 Recurrence Relations and Derivatives 13.17 Continued Fractions

§13.16 Integral Representations

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Permalink:: http://dlmf.nist.gov/13.16
See also:: Annotations for Ch.13

§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

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Keywords:: Whittaker functions, along the real line, hypergeometric function, integral representations, integrals, integrals of modified Bessel functions, over infinite intervals
Notes:: Combine §13.4(i) with (13.14.4) and (13.14.5).
Permalink:: http://dlmf.nist.gov/13.16.i
See also:: Annotations for §13.16 and Ch.13

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

13.16.1	$\displaystyle M_{\kappa,\mu}\left(z\right)$	$\displaystyle=\frac{\Gamma\left(1+2\mu\right)z^{\mu+\frac{1}{2}}2^{-2\mu}}{% \Gamma\left(\frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}{2}+\mu+\kappa% \right)}\*\int_{-1}^{1}e^{\frac{1}{2}zt}(1+t)^{\mu-\frac{1}{2}-\kappa}(1-t)^{% \mu-\frac{1}{2}+\kappa}\mathrm{d}t,$
$\Re\mu+\tfrac{1}{2}>\left\|\Re\kappa\right\|$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker 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: §13.29(iii) Permalink: http://dlmf.nist.gov/13.16.E1 Encodings: TeX, pMML, png See also: Annotations for §13.16(i), §13.16 and Ch.13
13.16.2	$\displaystyle M_{\kappa,\mu}\left(z\right)$	$\displaystyle=\frac{\Gamma\left(1+2\mu\right)z^{\lambda}}{\Gamma\left(1+2\mu-2% \lambda\right)\Gamma\left(2\lambda\right)}\*\int_{0}^{1}M_{\kappa-\lambda,\mu-% \lambda}\left(zt\right)e^{\frac{1}{2}z(t-1)}t^{\mu-\lambda-\frac{1}{2}}{(1-t)^% {2\lambda-1}}\mathrm{d}t,$
$\Re\mu+\tfrac{1}{2}>\Re\lambda>0$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker 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: §13.23(v) Permalink: http://dlmf.nist.gov/13.16.E2 Encodings: TeX, pMML, png See also: Annotations for §13.16(i), §13.16 and Ch.13

13.16.3	$\displaystyle\frac{1}{\Gamma\left(1+2\mu\right)}M_{\kappa,\mu}\left(z\right)$	$\displaystyle=\frac{\sqrt{z}e^{\frac{1}{2}z}}{\Gamma\left(\frac{1}{2}+\mu+% \kappa\right)}\int_{0}^{\infty}e^{-t}t^{\kappa-\frac{1}{2}}J_{2\mu}\left(2% \sqrt{zt}\right)\mathrm{d}t,$
$\Re\left(\kappa+\mu\right)+\tfrac{1}{2}>0$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker 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 Permalink: http://dlmf.nist.gov/13.16.E3 Encodings: TeX, pMML, png See also: Annotations for §13.16(i), §13.16 and Ch.13
13.16.4	$\displaystyle\frac{1}{\Gamma\left(1+2\mu\right)}M_{\kappa,\mu}\left(z\right)$	$\displaystyle=\frac{\sqrt{z}e^{-\frac{1}{2}z}}{\Gamma\left(\frac{1}{2}+\mu-% \kappa\right)}\*\int_{0}^{\infty}e^{-t}t^{-\kappa-\frac{1}{2}}I_{2\mu}\left(2% \sqrt{zt}\right)\mathrm{d}t,$
$\Re(\kappa-\mu)-\tfrac{1}{2}<0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\Re$ : real part and $z$ : complex variable Referenced by: Erratum (V1.0.5) for Equation (13.16.4) Permalink: http://dlmf.nist.gov/13.16.E4 Encodings: TeX, pMML, png Errata (effective with 1.0.5): Originally the condition for the validity of this equation was stated incorrectly as $\Re(\kappa-\mu)-\tfrac{1}{2}>0$ . The correct condition is $\Re(\kappa-\mu)-\tfrac{1}{2}<0$ . See also: Annotations for §13.16(i), §13.16 and Ch.13

13.16.5		$W_{\kappa,\mu}\left(z\right)=\frac{z^{\mu+\frac{1}{2}}2^{-2\mu}}{\Gamma\left(% \frac{1}{2}+\mu-\kappa\right)}\*\int_{1}^{\infty}e^{-\frac{1}{2}zt}(t-1)^{\mu-% \frac{1}{2}-\kappa}(t+1)^{\mu-\frac{1}{2}+\kappa}\mathrm{d}t,$
$\Re\mu+\tfrac{1}{2}>\Re\kappa$ , $\|\operatorname{ph}{z}\|<\frac{1}{2}\pi$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker 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, $\operatorname{ph}$ : phase, $\Re$ : real part and $z$ : complex variable Referenced by: §13.29(iii) Permalink: http://dlmf.nist.gov/13.16.E5 Encodings: TeX, pMML, png See also: Annotations for §13.16(i), §13.16 and Ch.13

13.16.6		$W_{\kappa,\mu}\left(z\right)=\frac{e^{-\frac{1}{2}z}z^{\kappa+1}}{\Gamma\left(% \frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}{2}-\mu-\kappa\right)}\*\int_% {0}^{\infty}\frac{W_{-\kappa,\mu}\left(t\right)e^{-\frac{1}{2}t}t^{-\kappa-1}}% {t+z}\mathrm{d}t,$
$\|\operatorname{ph}{z}\|<\pi$ , $\Re\left(\frac{1}{2}+\mu-\kappa\right)>\max\left(2\Re\mu,0\right)$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker 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, $\operatorname{ph}$ : phase, $\Re$ : real part and $z$ : complex variable Referenced by: §13.23(v) Permalink: http://dlmf.nist.gov/13.16.E6 Encodings: TeX, pMML, png See also: Annotations for §13.16(i), §13.16 and Ch.13

13.16.7		$W_{\kappa,\mu}\left(z\right)=\frac{(-1)^{n}e^{-\frac{1}{2}z}z^{\frac{1}{2}-\mu% -n}}{\Gamma\left(1+2\mu\right)\Gamma\left(\frac{1}{2}-\mu-\kappa\right)}\*\int% _{0}^{\infty}\frac{M_{-\kappa,\mu}\left(t\right)e^{-\frac{1}{2}t}t^{n+\mu-% \frac{1}{2}}}{t+z}\mathrm{d}t,$
$\|\operatorname{ph}z\|<\pi$ , $n=0,1,2,\dots$ , $-\Re\left(1+2\mu\right)<n<\left\|\Re\mu\right\|+\Re\kappa<\tfrac{1}{2}$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker 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, $\operatorname{ph}$ : phase, $\Re$ : real part, $n$ : nonnegative integer and $z$ : complex variable Referenced by: §13.23(v) Permalink: http://dlmf.nist.gov/13.16.E7 Encodings: TeX, pMML, png See also: Annotations for §13.16(i), §13.16 and Ch.13

13.16.8		$W_{\kappa,\mu}\left(z\right)=\frac{2\sqrt{z}e^{-\frac{1}{2}z}}{\Gamma\left(% \frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}{2}-\mu-\kappa\right)}\*\int_% {0}^{\infty}e^{-t}t^{-\kappa-\frac{1}{2}}K_{2\mu}\left(2\sqrt{zt}\right)% \mathrm{d}t,$
$\Re\left(\mu-\kappa\right)+\tfrac{1}{2}>0$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\Re$ : real part and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.16.E8 Encodings: TeX, pMML, png See also: Annotations for §13.16(i), §13.16 and Ch.13

13.16.9		$W_{\kappa,\mu}\left(z\right)=e^{-\frac{1}{2}z}z^{\kappa+c}\*\int_{0}^{\infty}e% ^{-zt}t^{c-1}{{}_{2}{\mathbf{F}}_{1}}\left({\tfrac{1}{2}+\mu-\kappa,\tfrac{1}{% 2}-\mu-\kappa\atop c};-t\right)\mathrm{d}t,$
$\|\operatorname{ph}{z}\|<\frac{1}{2}\pi$ ,
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker 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, ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ 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, $\operatorname{ph}$ : phase and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.16.E9 Encodings: TeX, pMML, png See also: Annotations for §13.16(i), §13.16 and Ch.13

where $c$ is arbitrary, $\Re c>0$ .

§13.16(ii) Contour Integrals

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Keywords:: Whittaker functions, contour integrals, integral representations
Permalink:: http://dlmf.nist.gov/13.16.ii
See also:: Annotations for §13.16 and Ch.13

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

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Keywords:: Mellin–Barnes type, Whittaker functions, integral representations
Notes:: See Buchholz (1969, §5.4), Erdélyi et al. (1953a, §6.11), and Slater (1960, Chapter 3).
Referenced by:: §13.16(ii)
Permalink:: http://dlmf.nist.gov/13.16.iii
See also:: Annotations for §13.16 and Ch.13

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

13.16.10		$\frac{1}{\Gamma\left(1+2\mu\right)}M_{\kappa,\mu}\left(e^{\pm\pi\mathrm{i}}z% \right)=\frac{e^{\frac{1}{2}z\pm(\frac{1}{2}+\mu)\pi\mathrm{i}}}{2\pi\mathrm{i% }\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}\*\int_{-\mathrm{i}\infty}^{\mathrm% {i}\infty}\frac{\Gamma\left(t-\kappa\right)\Gamma\left(\frac{1}{2}+\mu-t\right% )}{\Gamma\left(\frac{1}{2}+\mu+t\right)}z^{t}\mathrm{d}t,$
$\|\operatorname{ph}{z}\|<\tfrac{1}{2}\pi$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker 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, $\operatorname{ph}$ : phase and $z$ : complex variable Keywords: Mellin transform Permalink: http://dlmf.nist.gov/13.16.E10 Encodings: TeX, pMML, png See also: Annotations for §13.16(iii), §13.16 and Ch.13

where the contour of integration separates the poles of $\Gamma\left(t-\kappa\right)$ from those of $\Gamma\left(\frac{1}{2}+\mu-t\right)$ .

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

13.16.11		$W_{\kappa,\mu}\left(z\right)=\frac{e^{-\frac{1}{2}z}}{2\pi\mathrm{i}}\*\int_{-% \mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(\frac{1}{2}+\mu+t\right)% \Gamma\left(\frac{1}{2}-\mu+t\right)\Gamma\left(-\kappa-t\right)}{\Gamma\left(% \frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}{2}-\mu-\kappa\right)}z^{-t}% \mathrm{d}t,$
$\|\operatorname{ph}{z}\|<\tfrac{3}{2}\pi$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker 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, $\operatorname{ph}$ : phase and $z$ : complex variable Keywords: Mellin transform Permalink: http://dlmf.nist.gov/13.16.E11 Encodings: TeX, pMML, png See also: Annotations for §13.16(iii), §13.16 and Ch.13

where the contour of integration separates the poles of $\Gamma\left(\frac{1}{2}+\mu+t\right)\Gamma\left(\frac{1}{2}-\mu+t\right)$ from those of $\Gamma\left(-\kappa-t\right)$ .

13.16.12		$W_{\kappa,\mu}\left(z\right)=\frac{e^{\frac{1}{2}z}}{2\pi\mathrm{i}}\int_{-% \mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(\frac{1}{2}+\mu+t\right)% \Gamma\left(\frac{1}{2}-\mu+t\right)}{\Gamma\left(1-\kappa+t\right)}z^{-t}% \mathrm{d}t,$
$\|\operatorname{ph}{z}\|<\tfrac{1}{2}\pi$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker 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, $\operatorname{ph}$ : phase and $z$ : complex variable Keywords: Mellin transform Permalink: http://dlmf.nist.gov/13.16.E12 Encodings: TeX, pMML, png See also: Annotations for §13.16(iii), §13.16 and Ch.13

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

© 2010–2020 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.1.0; Release date 2020-12-15. A printed companion is available. 13.15 Recurrence Relations and Derivatives 13.17 Continued Fractions

§13.16 Integral Representations

Contents

§13.16(i) Integrals Along the Real Line

§13.16(ii) Contour Integrals

§13.16(iii) Mellin–Barnes Integrals