for confluent hypergeometric functions

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11: 13.1 Special Notation

… ►The main functions treated in this chapter are the Kummer functions $M(a,b,z)$ and $U(a,b,z)$, Olver’s function $𝐌(a,b,z)$, and the Whittaker functions $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$. … ►

12: 13.10 Integrals

… ►

13.10.6 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt-t^2}{t}^{2b-2}𝐌(a,b,t^2)dt=\frac{1}{2}{\pi }^{-\frac{1}{2}}\mathrm{\Gamma }\left(b-\frac{1}{2}\right)U(b-\frac{1}{2},a+\frac{1}{2},\frac{1}{4}{z}^2),$$ $\mathrm{\Re }b>\frac{1}{2}$, $\mathrm{\Re }z>0$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(ii), §13.10 and Ch.13

… ►

13.10.10 $${\int }_0^{\mathrm{\infty }}{t}^{\lambda -1}𝐌(a,b,-t)dt=\frac{\mathrm{\Gamma }\left(\lambda \right)\mathrm{\Gamma }\left(a-\lambda \right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b-\lambda \right)},$$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/13.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(iii), §13.10 and Ch.13

… ►

13.10.12 $${\int }_0^{\mathrm{\infty }}\cos \left(2xt\right)𝐌(a,b,-t^2)dt=\frac{\sqrt{\pi }}{2\mathrm{\Gamma }\left(a\right)}{x}^{2a-1}{\mathrm{e}}^{-x^2}U(b-\frac{1}{2},a+\frac{1}{2},x^2),$$ $\mathrm{\Re }a>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.10.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(iv), §13.10 and Ch.13

… ►

13.10.14 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{\frac{1}{2}\nu }𝐌(a,b,t)J_{\nu }\left(2\sqrt{xt}\right)dt=\frac{x^{\frac{1}{2}\nu }{\mathrm{e}}^{-x}}{\mathrm{\Gamma }\left(b-a\right)}U(a,a-b+\nu +2,x),$$ $x>0$, ,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable

Referenced by:

§13.10(v)

Permalink:

http://dlmf.nist.gov/13.10.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(v), §13.10 and Ch.13

… ►

13.10.16 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{\frac{1}{2}\nu }U(a,b,t)J_{\nu }\left(2\sqrt{xt}\right)dt=\mathrm{\Gamma }\left(\nu -b+2\right)x^{\frac{1}{2}\nu }{\mathrm{e}}^{-x}𝐌(a,a-b+\nu +2,x),$$ $x>0$, .

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable

Referenced by:

§13.10(v)

Permalink:

http://dlmf.nist.gov/13.10.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(v), §13.10 and Ch.13

…

13: 13.14 Definitions and Basic Properties

… ►

13.14.3 $$W_{\kappa ,\mu }\left(z\right)={\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}+\mu }U(\frac{1}{2}+\mu -\kappa ,1+2\mu ,z),$$

ⓘ

Defines:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

A&S Ref:

13.1.33

Referenced by:

§13.14(i), §13.16(ii), §13.19, §13.20(i), §13.22, §13.4(iii), §18.34(i), §18.39(i)

Permalink:

http://dlmf.nist.gov/13.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13

… ►Except when $z=0$, each branch of the functions $M_{\kappa ,\mu }\left(z\right)/\mathrm{\Gamma }\left(2\mu +1\right)$ and $W_{\kappa ,\mu }\left(z\right)$ is entire in $\kappa$ and $\mu$. … ►

13.14.26 $$𝒲\left\{M_{\kappa ,\mu }\left(z\right),W_{\kappa ,\mu }\left(z\right)\right\}=-\frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $𝒲$: Wronskian and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.14.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(vi), §13.14 and Ch.13

… ►

13.14.28 $$𝒲\left\{M_{\kappa ,-\mu }\left(z\right),W_{\kappa ,\mu }\left(z\right)\right\}=-\frac{\mathrm{\Gamma }\left(1-2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\kappa \right)},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $𝒲$: Wronskian and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.14.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(vi), §13.14 and Ch.13

… ►

13.14.33 $$W_{\kappa ,\mu }\left(z\right)=\frac{\mathrm{\Gamma }\left(-2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\kappa \right)}{M}_{\kappa ,\mu }\left(z\right)+\frac{\mathrm{\Gamma }\left(2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{M}_{\kappa ,-\mu }\left(z\right).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.1.34

Referenced by:

§13.14(i), §13.14(vii)

Permalink:

http://dlmf.nist.gov/13.14.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(vii), §13.14 and Ch.13

14: 13.23 Integrals

… ►

13.23.10 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}t}{t}^{\frac{1}{2}(\nu -1)-\mu }{M}_{\kappa ,\mu }\left(t\right)J_{\nu }\left(2\sqrt{xt}\right)dt=\frac{{\mathrm{e}}^{-\frac{1}{2}x}{x}^{\frac{1}{2}(\kappa +\mu -\frac{3}{2})}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)}{W}_{\frac{1}{2}(\kappa -3\mu +\nu +\frac{1}{2}),\frac{1}{2}(\kappa +\mu -\nu -\frac{1}{2})}\left(x\right),$$ $x>0$, .

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.23.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(iii), §13.23 and Ch.13

… ►

13.23.12 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}t}{t}^{\frac{1}{2}(\nu -1)-\mu }{W}_{\kappa ,\mu }\left(t\right)J_{\nu }\left(2\sqrt{xt}\right)dt=\frac{\mathrm{\Gamma }\left(\nu -2\mu +1\right)}{\mathrm{\Gamma }\left(\frac{3}{2}-\mu -\kappa +\nu \right)}{\mathrm{e}}^{-\frac{1}{2}x}{x}^{\frac{1}{2}(\mu +\kappa -\frac{3}{2})}{M}_{\frac{1}{2}(\kappa -3\mu +\nu +\frac{1}{2}),\frac{1}{2}(\nu -\mu -\kappa +\frac{1}{2})}\left(x\right),$$ $x>0$, .

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.23.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(iii), §13.23 and Ch.13

… ►

13.23.13 $$g(\mu )=\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{\int }_0^{\mathrm{\infty }}f(x)x^{-\frac{3}{2}}{M}_{\kappa ,\mu }\left(x\right)dx,$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\int$: integral and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.23.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(iv), §13.23 and Ch.13

►

13.23.14 $$f(x)=\frac{1}{\pi \mathrm{i}\sqrt{x}}{\int }_{{\mu }_1-\mathrm{i}\mathrm{\infty }}^{{\mu }_1+\mathrm{i}\mathrm{\infty }}\mu g(\mu )\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)W_{\kappa ,\mu }\left(x\right)d\mu .$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.23.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(iv), §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 (2000, §7.6), and Prudnikov et al. (1990, §§1.13, 1.14, 2.19, 4.2.2). …

15: 10.16 Relations to Other Functions

… ►

Confluent Hypergeometric Functions

►

10.16.5 $$J_{\nu }\left(z\right)=\frac{{(\frac{1}{2}z)}^{\nu }{\mathrm{e}}^{\mp \mathrm{i}z}}{\mathrm{\Gamma }\left(\nu +1\right)}M(\nu +\frac{1}{2},2\nu +1,\pm 2\mathrm{i}z),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.69 (modified)

Referenced by:

§10.16, §10.39

Permalink:

http://dlmf.nist.gov/10.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.16, §10.16 and Ch.10

… ►For the functions $M$ and $U$ see §13.2(i). ►

10.16.7 $$J_{\nu }\left(z\right)=\frac{{\mathrm{e}}^{\mp (2\nu +1)\pi \mathrm{i}/4}}{2^{2\nu }\mathrm{\Gamma }\left(\nu +1\right)}{(2z)}^{-\frac{1}{2}}{M}_{0,\nu }\left(\pm 2\mathrm{i}z\right),$$ $2\nu \ne -1,-2,-3,\mathrm{\dots }$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.16, §10.16

Permalink:

http://dlmf.nist.gov/10.16.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.16, §10.16 and Ch.10

… ►For the functions $M_{0,\nu }$ and $W_{0,\nu }$ see §13.14(i). …

16: 13.4 Integral Representations

… ►

13.4.1 $$𝐌(a,b,z)=\frac{1}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b-a\right)}{\int }_0^1{\mathrm{e}}^{zt}{t}^{a-1}{(1-t)}^{b-a-1}dt,$$ $\mathrm{\Re }b>\mathrm{\Re }a>0$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

13.2.1 (in different form)

Referenced by:

§13.10(v), §13.29(iii)

Permalink:

http://dlmf.nist.gov/13.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

►

13.4.2 $$𝐌(a,b,z)=\frac{1}{\mathrm{\Gamma }\left(b-c\right)}{\int }_0^1𝐌(a,c,zt)t^{c-1}{(1-t)}^{b-c-1}dt,$$ $\mathrm{\Re }b>\mathrm{\Re }c>0$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable and $c$: arbitrary

Referenced by:

§13.10(vi)

Permalink:

http://dlmf.nist.gov/13.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

►

13.4.3 $$𝐌(a,b,-z)=\frac{z^{\frac{1}{2}-\frac{1}{2}b}}{\mathrm{\Gamma }\left(a\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{a-\frac{1}{2}b-\frac{1}{2}}{J}_{b-1}\left(2\sqrt{zt}\right)dt,$$ $\mathrm{\Re }a>0$.

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§13.10(v)

Permalink:

http://dlmf.nist.gov/13.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

… ►

13.4.4 $$U(a,b,z)=\frac{1}{\mathrm{\Gamma }\left(a\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{a-1}{(1+t)}^{b-a-1}dt,$$ $\mathrm{\Re }a>0$, ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

13.2.5

Referenced by:

§13.10(v), §13.29(iii), §7.11, (9.5.8)

Permalink:

http://dlmf.nist.gov/13.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

… ►

13.4.6 $$U(a,b,z)=\frac{{(-1)}^n{z}^{1-b-n}}{\mathrm{\Gamma }\left(1+a-b\right)}{\int }_0^{\mathrm{\infty }}\frac{𝐌(b-a,b,t){\mathrm{e}}^{-t}{t}^{b+n-1}}{t+z}dt,$$ , $n=0,1,2,\mathrm{\dots }$, ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.10(vi)

Permalink:

http://dlmf.nist.gov/13.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

…

17: 13.16 Integral Representations

… ►

13.16.1 $$M_{\kappa ,\mu }\left(z\right)=\frac{\mathrm{\Gamma }\left(1+2\mu \right)z^{\mu +\frac{1}{2}}{2}^{-2\mu }}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)}{\int }_{-1}^1{\mathrm{e}}^{\frac{1}{2}zt}{(1+t)}^{\mu -\frac{1}{2}-\kappa }{(1-t)}^{\mu -\frac{1}{2}+\kappa }dt,$$ $\mathrm{\Re }\mu +\frac{1}{2}>\left|\mathrm{\Re }\kappa \right|$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§13.29(iii)

Permalink:

http://dlmf.nist.gov/13.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(i), §13.16 and Ch.13

►

13.16.2 $$M_{\kappa ,\mu }\left(z\right)=\frac{\mathrm{\Gamma }\left(1+2\mu \right)z^{\lambda }}{\mathrm{\Gamma }\left(1+2\mu -2\lambda \right)\mathrm{\Gamma }\left(2\lambda \right)}{\int }_0^1{M}_{\kappa -\lambda ,\mu -\lambda }\left(zt\right){\mathrm{e}}^{\frac{1}{2}z(t-1)}{t}^{\mu -\lambda -\frac{1}{2}}{(1-t)}^{2\lambda -1}dt,$$ $\mathrm{\Re }\mu +\frac{1}{2}>\mathrm{\Re }\lambda >0$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§13.23(v)

Permalink:

http://dlmf.nist.gov/13.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(i), §13.16 and Ch.13

►

13.16.3 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{M}_{\kappa ,\mu }\left(z\right)=\frac{\sqrt{z}{\mathrm{e}}^{\frac{1}{2}z}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{\kappa -\frac{1}{2}}{J}_{2\mu }\left(2\sqrt{zt}\right)dt,$$ $\mathrm{\Re }\left(\kappa +\mu \right)+\frac{1}{2}>0$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.16.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(i), §13.16 and Ch.13

… ►

13.16.7 $$W_{\kappa ,\mu }\left(z\right)=\frac{{(-1)}^n{\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}-\mu -n}}{\mathrm{\Gamma }\left(1+2\mu \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\kappa \right)}{\int }_0^{\mathrm{\infty }}\frac{M_{-\kappa ,\mu }\left(t\right){\mathrm{e}}^{-\frac{1}{2}t}{t}^{n+\mu -\frac{1}{2}}}{t+z}dt,$$ , $n=0,1,2,\mathrm{\dots }$, ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.23(v)

Permalink:

http://dlmf.nist.gov/13.16.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(i), §13.16 and Ch.13

►

13.16.8 $$W_{\kappa ,\mu }\left(z\right)=\frac{2\sqrt{z}{\mathrm{e}}^{-\frac{1}{2}z}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\kappa \right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{-\kappa -\frac{1}{2}}{K}_{2\mu }\left(2\sqrt{zt}\right)dt,$$ $\mathrm{\Re }\left(\mu -\kappa \right)+\frac{1}{2}>0$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.16.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(i), §13.16 and Ch.13

…

18: 12.18 Methods of Computation

… ►Because PCFs are special cases of confluent hypergeometric functions, the methods of computation described in §13.29 are applicable to PCFs. …

19: 10.39 Relations to Other Functions

… ►

Confluent Hypergeometric Functions

►

10.39.5 $$I_{\nu }\left(z\right)=\frac{{(\frac{1}{2}z)}^{\nu }{\mathrm{e}}^{\pm z}}{\mathrm{\Gamma }\left(\nu +1\right)}M(\nu +\frac{1}{2},2\nu +1,\mp 2z),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.39

Permalink:

http://dlmf.nist.gov/10.39.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

… ►

10.39.7 $$I_{\nu }\left(z\right)=\frac{{(2z)}^{-\frac{1}{2}}{M}_{0,\nu }\left(2z\right)}{2^{2\nu }\mathrm{\Gamma }\left(\nu +1\right)},$$ $2\nu \ne -1,-2,-3,\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.47

Referenced by:

(9.6.23), (9.6.24)

Permalink:

http://dlmf.nist.gov/10.39.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

►

10.39.8 $$K_{\nu }\left(z\right)={\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{W}_{0,\nu }\left(2z\right).$$

ⓘ

Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.48

Referenced by:

(9.6.21), (9.6.22)

Permalink:

http://dlmf.nist.gov/10.39.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

►For the functions $M$, $U$, $M_{0,\nu }$, and $W_{0,\nu }$ see §§13.2(i) and 13.14(i). …

20: 35.10 Methods of Computation

… ►See Yan (1992) for the ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions of matrix argument in the case $m=2$, and Bingham et al. (1992) for Monte Carlo simulation on $𝐎(m)$ applied to a generalization of the integral (35.5.8). …