13 Confluent Hypergeometric Functions Whittaker Functions 13.22 Zeros 13.24 Series

§13.23 Integrals

Referenced by:: ¶ ‣ §12.12
Permalink:: http://dlmf.nist.gov/13.23

§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

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).
Keywords:: Whittaker functions, integrals
Permalink:: http://dlmf.nist.gov/13.23.i

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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : differential of , : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and : complex variable
Referenced by:: §13.23(i)
Permalink:: http://dlmf.nist.gov/13.23.E1
Encodings:: TeX, pMML, png

13.23.2 $\int_{{0}}^{{\infty}}e^{{-zt}}t^{{\mu-\frac{1}{2}}}\mathop{M_{{\kappa,\mu}}\/}% \nolimits\!\left(t\right)dt=\mathop{\Gamma\/}\nolimits\!\left(2\mu+1\right)% \left(z+\tfrac{1}{2}\right)^{{-\kappa-\mu-\frac{1}{2}}}\*\left(z-\tfrac{1}{2}% \right)^{{\kappa-\mu-\frac{1}{2}}},$ $\realpart{\mu}>-\tfrac{1}{2}$ , $\realpart{z}>\tfrac{1}{2}$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : differential of , : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and : complex variable
Permalink:: http://dlmf.nist.gov/13.23.E2
Encodings:: TeX, pMML, png

13.23.3 $\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(1+2\mu\right)}\int_{{0}}^{{\infty}}% e^{{-\frac{1}{2}t}}t^{{\nu-1}}\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(t% \right)dt=\frac{\mathop{\Gamma\/}\nolimits\!\left(\mu+\nu+\frac{1}{2}\right)% \mathop{\Gamma\/}\nolimits\!\left(\kappa-\nu\right)}{\mathop{\Gamma\/}% \nolimits\!\left(\frac{1}{2}+\mu+\kappa\right)\mathop{\Gamma\/}\nolimits\!% \left(\frac{1}{2}+\mu-\nu\right)},$ $-\tfrac{1}{2}-\realpart{\mu}<\realpart{\nu}<\realpart{\kappa}$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : differential of , : base of exponential function, $\int$ : integral and $\realpart{}$ : real part
Referenced by:: §13.23(i)
Permalink:: http://dlmf.nist.gov/13.23.E3
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : differential of , : 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, $\realpart{}$ : real part and : complex variable
Referenced by:: §13.23(i)
Permalink:: http://dlmf.nist.gov/13.23.E4
Encodings:: TeX, pMML, png

13.23.5 $\int_{{0}}^{{\infty}}e^{{\frac{1}{2}t}}t^{{\nu-1}}\mathop{W_{{\kappa,\mu}}\/}% \nolimits\!\left(t\right)dt=\frac{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2% }+\mu+\nu\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\mu+\nu\right)% \mathop{\Gamma\/}\nolimits\!\left(-\kappa-\nu\right)}{\mathop{\Gamma\/}% \nolimits\!\left(\frac{1}{2}+\mu-\kappa\right)\mathop{\Gamma\/}\nolimits\!% \left(\frac{1}{2}-\mu-\kappa\right)},$ $|\realpart{\mu}|-\tfrac{1}{2}<\realpart{\nu}<-\realpart{\kappa}$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : differential of , : base of exponential function, $\int$ : integral and $\realpart{}$ : real part
Referenced by:: §13.23(i)
Permalink:: http://dlmf.nist.gov/13.23.E5
Encodings:: TeX, pMML, png

13.23.6 $\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(1+2\mu\right)2\pi i}\int_{{-\infty}% }^{{(0+)}}e^{{zt+\frac{1}{2}t^{{-1}}}}t^{{\kappa}}\mathop{M_{{\kappa,\mu}}\/}% \nolimits\!\left(t^{{-1}}\right)dt=\frac{z^{{-\kappa-\frac{1}{2}}}}{\mathop{% \Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu-\kappa\right)}\mathop{I_{{2\mu}}\/}% \nolimits\!\left(2\sqrt{z}\right),$ $\realpart{z}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : differential of , : base of exponential function, $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part and : complex variable
Permalink:: http://dlmf.nist.gov/13.23.E6
Encodings:: TeX, pMML, png

13.23.7 $\frac{1}{2\pi i}\int_{{-\infty}}^{{(0+)}}e^{{zt+\frac{1}{2}t^{{-1}}}}t^{{% \kappa}}\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(t^{{-1}}\right)dt=\frac{2z% ^{{-\kappa-\frac{1}{2}}}}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu-% \kappa\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\mu-\kappa\right)}% \mathop{K_{{2\mu}}\/}\nolimits\!\left(2\sqrt{z}\right),$ $\realpart{z}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : differential of , : base of exponential function, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part and : complex variable
Permalink:: http://dlmf.nist.gov/13.23.E7
Encodings:: TeX, pMML, png

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

Notes:: See Erdélyi et al. (1953a, §6.15.2).
Keywords:: Whittaker functions, integrals
Permalink:: http://dlmf.nist.gov/13.23.ii

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

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, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and : real variable
Permalink:: http://dlmf.nist.gov/13.23.E8
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).

§13.23(iii) Hankel Transforms

Notes:: Combine §13.10(v) with (13.14.4), (13.14.5).
Keywords:: Whittaker functions, integrals
Permalink:: http://dlmf.nist.gov/13.23.iii

For the notation see §10.2(ii).

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

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

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, : differential of , : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and : real variable
Permalink:: http://dlmf.nist.gov/13.23.E9
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : differential of , : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and : real variable
Permalink:: http://dlmf.nist.gov/13.23.E10
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : differential of , : base of exponential function, $\int$ : integral, : open interval, $\realpart{}$ : real part and : real variable
Permalink:: http://dlmf.nist.gov/13.23.E11
Encodings:: TeX, pMML, png

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

, $\max(2\realpart{\mu}-1,-1)<\realpart{\nu}$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : differential of , : base of exponential function, $\int$ : integral, : open interval, $\realpart{}$ : real part and : real variable
Permalink:: http://dlmf.nist.gov/13.23.E12
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).

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

Notes:: See Snow (1952, Chapter XI).
Keywords:: Whittaker functions, integral transforms
Referenced by:: §13.10(vi)
Permalink:: http://dlmf.nist.gov/13.23.iv

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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : differential of , $\int$ : integral and : real variable
Permalink:: http://dlmf.nist.gov/13.23.E13
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : differential of , $\int$ : integral and : real variable
Permalink:: http://dlmf.nist.gov/13.23.E14
Encodings:: TeX, pMML, png

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

Keywords:: Whittaker functions, integrals
Permalink:: http://dlmf.nist.gov/13.23.v

Additional integrals involving confluent hypergeometric functions can be found in Apelblat (1983, pp. 388–392), Erdélyi et al. (1954b), Gradshteyn and Ryzhik (2000, §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).