13 Confluent Hypergeometric Functions Whittaker Functions 13.22 Zeros 13.24 Series

§13.23 Integrals

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Referenced by:: §12.12
Permalink:: http://dlmf.nist.gov/13.23
See also:: Annotations for Ch.13

§13.23(i) Laplace and Mellin Transforms
§13.23(ii) Fourier Transforms
§13.23(iii) Hankel Transforms
§13.23(iv) Integral Transforms in terms of Whittaker Functions
§13.23(v) Other Integrals

§13.23(i) Laplace and Mellin Transforms

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Keywords:: Laplace transforms, Mellin transforms, Whittaker functions, integrals
Notes:: See Buchholz (1969, §10), Erdélyi et al. (1953a, §6.10), and Slater (1960, Chapter 3). (13.23.3) and (13.23.5) can also be derived as limiting forms of (13.23.1) and (13.23.4), respectively; compare (15.4.20).
Permalink:: http://dlmf.nist.gov/13.23.i
Clarification (effective with 1.2.1):: Just below (13.23.7), we added “For the particular loop contour, see Figure 5.9.1.”
See also:: Annotations for §13.23 and Ch.13

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

13.23.1 $\int_{0}^{\infty}e^{-zt}t^{\nu-1}M_{\kappa,\mu}\left(t\right)\,\mathrm{d}t=% \frac{\Gamma\left(\mu+\nu+\tfrac{1}{2}\right)}{\left(z+\frac{1}{2}\right)^{\mu% +\nu+\frac{1}{2}}}\*{{}_{2}F_{1}}\left({\tfrac{1}{2}+\mu-\kappa,\tfrac{1}{2}+% \mu+\nu\atop 1+2\mu};\frac{1}{z+\frac{1}{2}}\right),$ $\Re\mu+\nu+\tfrac{1}{2}>0$ , $\Re z>\tfrac{1}{2}$ .

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part and : complex variable
Keywords:: Laplace transform, Mellin transform
Referenced by:: §13.23(i)
Permalink:: http://dlmf.nist.gov/13.23.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §13.23(i), §13.23 and Ch.13

13.23.2 $\int_{0}^{\infty}e^{-zt}t^{\mu-\frac{1}{2}}M_{\kappa,\mu}\left(t\right)\,% \mathrm{d}t=\Gamma\left(2\mu+1\right)\left(z+\tfrac{1}{2}\right)^{-\kappa-\mu-% \frac{1}{2}}\*\left(z-\tfrac{1}{2}\right)^{\kappa-\mu-\frac{1}{2}},$ $\Re\mu>-\tfrac{1}{2}$ , $\Re z>\tfrac{1}{2}$ ,

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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 , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part and : complex variable
Keywords:: Laplace transform, Mellin transform
Permalink:: http://dlmf.nist.gov/13.23.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §13.23(i), §13.23 and Ch.13

13.23.3 $\frac{1}{\Gamma\left(1+2\mu\right)}\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\nu-1}% M_{\kappa,\mu}\left(t\right)\,\mathrm{d}t=\frac{\Gamma\left(\mu+\nu+\frac{1}{2% }\right)\Gamma\left(\kappa-\nu\right)}{\Gamma\left(\frac{1}{2}+\mu+\kappa% \right)\Gamma\left(\frac{1}{2}+\mu-\nu\right)},$ $-\tfrac{1}{2}-\Re\mu<\Re\nu<\Re\kappa$ .

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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 , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $\Re$ : real part
Keywords:: Laplace transform, Mellin transform
Referenced by:: §13.23(i)
Permalink:: http://dlmf.nist.gov/13.23.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §13.23(i), §13.23 and Ch.13

13.23.4 $\int_{0}^{\infty}e^{-zt}t^{\nu-1}W_{\kappa,\mu}\left(t\right)\,\mathrm{d}t=% \Gamma\left(\tfrac{1}{2}+\mu+\nu\right)\Gamma\left(\tfrac{1}{2}-\mu+\nu\right)% \*{{}_{2}{\mathbf{F}}_{1}}\left({\tfrac{1}{2}-\mu+\nu,\tfrac{1}{2}+\mu+\nu% \atop\nu-\kappa+1};\tfrac{1}{2}-z\right),$ $\Re\left(\nu+\tfrac{1}{2}\right)>|\Re\mu|$ , $\Re z>-\tfrac{1}{2}$ ,

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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 , $\mathrm{e}$ : 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 : complex variable
Keywords:: Laplace transform, Mellin transform
Referenced by:: §13.23(i)
Permalink:: http://dlmf.nist.gov/13.23.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §13.23(i), §13.23 and Ch.13

13.23.5 $\int_{0}^{\infty}e^{\frac{1}{2}t}t^{\nu-1}W_{\kappa,\mu}\left(t\right)\,% \mathrm{d}t=\frac{\Gamma\left(\frac{1}{2}+\mu+\nu\right)\Gamma\left(\frac{1}{2% }-\mu+\nu\right)\Gamma\left(-\kappa-\nu\right)}{\Gamma\left(\frac{1}{2}+\mu-% \kappa\right)\Gamma\left(\frac{1}{2}-\mu-\kappa\right)},$ $|\Re\mu|-\tfrac{1}{2}<\Re\nu<-\Re\kappa$ .

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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 , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $\Re$ : real part
Keywords:: Mellin transform
Referenced by:: §13.23(i)
Permalink:: http://dlmf.nist.gov/13.23.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §13.23(i), §13.23 and Ch.13

13.23.6 $\frac{1}{\Gamma\left(1+2\mu\right)2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{zt+% \frac{1}{2}t^{-1}}t^{\kappa}M_{\kappa,\mu}\left(t^{-1}\right)\,\mathrm{d}t=% \frac{z^{-\kappa-\frac{1}{2}}}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}I_{2% \mu}\left(2\sqrt{z}\right),$ $\Re z>0$ .

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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 , $\mathrm{e}$ : 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 and : complex variable
Keywords:: Mellin transform
Permalink:: http://dlmf.nist.gov/13.23.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §13.23(i), §13.23 and Ch.13

13.23.7 $\frac{1}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{zt+\frac{1}{2}t^{-1}}t^{\kappa% }W_{\kappa,\mu}\left(t^{-1}\right)\,\mathrm{d}t=\frac{2z^{-\kappa-\frac{1}{2}}% }{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}{2}-\mu-\kappa% \right)}K_{2\mu}\left(2\sqrt{z}\right),$ $\Re z>0$ .

For the particular loop contour, see Figure 5.9.1.

For additional Laplace and Mellin transforms see Erdélyi et al. (1954a, §§4.22, 5.20, 6.9, 7.5), Marichev (1983, pp. 283–287), Oberhettinger and Badii (1973, §1.17), Oberhettinger (1974, §§1.13, 2.8), 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).

§13.23(ii) Fourier Transforms

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Keywords:: Fourier transforms, Whittaker functions, integrals
Notes:: See Erdélyi et al. (1953a, §6.15.2).
Permalink:: http://dlmf.nist.gov/13.23.ii
See also:: Annotations for §13.23 and Ch.13

13.23.8 $\frac{1}{\Gamma\left(1+2\mu\right)}\int_{0}^{\infty}\cos\left(2xt\right)e^{-% \frac{1}{2}t^{2}}t^{-2\mu-1}M_{\kappa,\mu}\left(t^{2}\right)\,\mathrm{d}t=% \frac{\sqrt{\pi}e^{-\frac{1}{2}x^{2}}x^{\mu+\kappa-1}}{2\Gamma\left(\frac{1}{2% }+\mu+\kappa\right)}W_{\frac{1}{2}\kappa-\frac{3}{2}\mu,\frac{1}{2}\kappa+% \frac{1}{2}\mu}\left(x^{2}\right),$ $\Re\left(\kappa+\mu\right)>-\tfrac{1}{2}$ .

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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, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part and : real variable
Permalink:: http://dlmf.nist.gov/13.23.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §13.23(ii), §13.23 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).

§13.23(iii) Hankel Transforms

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Keywords:: Hankel transforms, Whittaker functions, integrals
Notes:: Combine §13.10(v) with (13.14.4), (13.14.5).
Permalink:: http://dlmf.nist.gov/13.23.iii
See also:: Annotations for §13.23 and Ch.13

For the notation see §10.2(ii).

13.23.9 $\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\mu-\frac{1}{2}(\nu+1)}M_{\kappa,\mu}% \left(t\right)J_{\nu}\left(2\sqrt{xt}\right)\,\mathrm{d}t=\frac{\Gamma\left(1+% 2\mu\right)}{\Gamma\left(\frac{1}{2}-\mu+\kappa+\nu\right)}\*e^{-\frac{1}{2}x}% x^{\frac{1}{2}(\kappa-\mu-\frac{3}{2})}\*M_{\frac{1}{2}(\kappa+3\mu-\nu+\frac{% 1}{2}),\frac{1}{2}(\kappa-\mu+\nu-\frac{1}{2})}\left(x\right),$

, $-\tfrac{1}{2}<\Re\mu<\Re\left(\kappa+\tfrac{1}{2}\nu\right)+\tfrac{3}{4}$ ,

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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 , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part and : real variable
Permalink:: http://dlmf.nist.gov/13.23.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §13.23(iii), §13.23 and Ch.13

13.23.10 $\frac{1}{\Gamma\left(1+2\mu\right)}\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\frac{% 1}{2}(\nu-1)-\mu}M_{\kappa,\mu}\left(t\right)J_{\nu}\left(2\sqrt{xt}\right)\,% \mathrm{d}t=\frac{e^{-\frac{1}{2}x}x^{\frac{1}{2}(\kappa+\mu-\frac{3}{2})}}{% \Gamma\left(\frac{1}{2}+\mu+\kappa\right)}\*W_{\frac{1}{2}(\kappa-3\mu+\nu+% \frac{1}{2}),\frac{1}{2}(\kappa+\mu-\nu-\frac{1}{2})}\left(x\right),$

, $-1<\Re\nu<2\Re\left(\mu+\kappa\right)+\tfrac{1}{2}$ .

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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, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part and : real variable
Permalink:: http://dlmf.nist.gov/13.23.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §13.23(iii), §13.23 and Ch.13

13.23.11 $\int_{0}^{\infty}e^{\frac{1}{2}t}t^{\frac{1}{2}(\nu-1)-\mu}W_{\kappa,\mu}\left% (t\right)J_{\nu}\left(2\sqrt{xt}\right)\,\mathrm{d}t=\frac{\Gamma\left(\nu-2% \mu+1\right)}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}\*e^{\frac{1}{2}x}x^{% \frac{1}{2}(\mu-\kappa-\frac{3}{2})}\*W_{\frac{1}{2}(\kappa+3\mu-\nu-\frac{1}{% 2}),\frac{1}{2}(\kappa-\mu+\nu+\frac{1}{2})}\left(x\right),$

, $\max(2\Re\mu-1,-1)<\Re\nu<2\Re\mu-\kappa+\tfrac{3}{2}$ ,

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part and : real variable
Permalink:: http://dlmf.nist.gov/13.23.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §13.23(iii), §13.23 and Ch.13

13.23.12 $\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\frac{1}{2}(\nu-1)-\mu}W_{\kappa,\mu}% \left(t\right)J_{\nu}\left(2\sqrt{xt}\right)\,\mathrm{d}t=\frac{\Gamma\left(% \nu-2\mu+1\right)}{\Gamma\left(\frac{3}{2}-\mu-\kappa+\nu\right)}\*e^{-\frac{1% }{2}x}x^{\frac{1}{2}(\mu+\kappa-\frac{3}{2})}\*M_{\frac{1}{2}(\kappa-3\mu+\nu+% \frac{1}{2}),\frac{1}{2}(\nu-\mu-\kappa+\frac{1}{2})}\left(x\right),$

, $\max(2\Re\mu-1,-1)<\Re\nu$ .

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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, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part and : real variable
Permalink:: http://dlmf.nist.gov/13.23.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §13.23(iii), §13.23 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).

§13.23(iv) Integral Transforms in terms of Whittaker Functions

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Keywords:: Whittaker functions, in terms of Whittaker functions, integral transforms, integral transforms in terms of
Notes:: See Snow (1952, Chapter XI).
Referenced by:: §13.10(vi)
Permalink:: http://dlmf.nist.gov/13.23.iv
See also:: Annotations for §13.23 and Ch.13

Let be absolutely integrable on the interval for all positive , $f(x)=O\left(x^{\rho_{0}}\right)$ as $x\to 0+$ , and $f(x)=O\left(e^{-\rho_{1}x}\right)$ as $x\to+\infty$ , where $\rho_{1}>\frac{1}{2}$ . Then for $\mu$ in the half-plane $\Re\mu\geq\mu_{1}>\max\left(-\rho_{0},\Re\kappa-\frac{1}{2}\right)$

13.23.13 $g(\mu)=\frac{1}{\Gamma\left(1+2\mu\right)}\int_{0}^{\infty}f(x)x^{-\frac{3}{2}% }M_{\kappa,\mu}\left(x\right)\,\mathrm{d}x,$

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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 , $\int$ : integral and : real variable
Permalink:: http://dlmf.nist.gov/13.23.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §13.23(iv), §13.23 and Ch.13

13.23.14 $f(x)=\frac{1}{\pi\mathrm{i}\sqrt{x}}\int_{\mu_{1}-\mathrm{i}\infty}^{\mu_{1}+% \mathrm{i}\infty}\mu g(\mu)\Gamma\left(\tfrac{1}{2}+\mu-\kappa\right)W_{\kappa% ,\mu}\left(x\right)\,\mathrm{d}\mu.$

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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 , $\mathrm{i}$ : imaginary unit, $\int$ : integral and : real variable
Permalink:: http://dlmf.nist.gov/13.23.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §13.23(iv), §13.23 and Ch.13

For additional integral transforms see Magnus et al. (1966, p. 189), Prudnikov et al. (1992b, §§4.3.39–4.3.42), and Wimp (1964).

§13.23(v) Other Integrals

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Keywords:: Whittaker functions, compendia, integrals
Referenced by:: Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)
Permalink:: http://dlmf.nist.gov/13.23.v
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 Whittaker functions.
See also:: Annotations for §13.23 and Ch.13

Additional integrals involving confluent hypergeometric functions can be found in Apelblat (1983, pp. 388–392), Erdélyi et al. (1954b), Gradshteyn and Ryzhik (2015, §7.6), and Prudnikov et al. (1990, §§1.13, 1.14, 2.19, 4.2.2). See also (13.16.2), (13.16.6), (13.16.7). Generalized orthogonality integrals (33.14.13) and (33.14.15) can be expressed in terms of Whittaker functions via the definitions in that section.

13.22 Zeros 13.24 Series

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