Kummer functions

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31: 16.2 Definition and Analytic Properties

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§16.2(i) Generalized Hypergeometric Series

… ► … ►Unless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values. … ►

Polynomials

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§16.2(v) Behavior with Respect to Parameters

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32: 16.18 Special Cases

… ►The ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions introduced in Chapters 13 and 15, as well as the more general ${}_p{}^{}F_q^{}$ functions introduced in the present chapter, are all special cases of the Meijer $G$-function. …As a corollary, special cases of the ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer $G$-function. …

33: 9.6 Relations to Other Functions

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9.6.21 $$\mathrm{Ai}\left(z\right)=\frac{1}{2}{\pi }^{-1/2}{z}^{-1/4}{W}_{0,1/3}\left(2\zeta \right)=3^{-1/6}{\pi }^{-1/2}{\zeta }^{2/3}{\mathrm{e}}^{-\zeta }U(\frac{5}{6},\frac{5}{3},2\zeta ),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $U(a,b,z)$: Kummer 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, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $\zeta$: change of variable

Proof sketch:

Combine (9.6.2) with (10.39.8) and (10.39.6).

Referenced by:

(9.5.8)

Permalink:

http://dlmf.nist.gov/9.6.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(iii), §9.6 and Ch.9

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9.6.22 $${\mathrm{Ai}}^{\prime }\left(z\right)=-\frac{1}{2}{\pi }^{-1/2}{z}^{1/4}{W}_{0,2/3}\left(2\zeta \right)=-3^{1/6}{\pi }^{-1/2}{\zeta }^{4/3}{\mathrm{e}}^{-\zeta }U(\frac{7}{6},\frac{7}{3},2\zeta ),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $U(a,b,z)$: Kummer 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, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $\zeta$: change of variable

Proof sketch:

Combine (9.6.3) with (10.39.8) and (10.39.6).

Permalink:

http://dlmf.nist.gov/9.6.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(iii), §9.6 and Ch.9

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9.6.25 $$\mathrm{Bi}\left(z\right)=\frac{1}{3^{1/6}\mathrm{\Gamma }\left(\frac{2}{3}\right)}{\mathrm{e}}^{-\zeta }{}_1{}^{}F_1^{}(\frac{1}{6};\frac{1}{3};2\zeta )+\frac{3^{5/6}}{2^{2/3}\mathrm{\Gamma }\left(\frac{1}{3}\right)}{\zeta }^{2/3}{\mathrm{e}}^{-\zeta }{}_1{}^{}F_1^{}(\frac{5}{6};\frac{5}{3};2\zeta ),$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $\zeta$: change of variable

Proof sketch:

Combine (9.6.23) with (13.14.2) and refer to §13.1.

Permalink:

http://dlmf.nist.gov/9.6.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(iii), §9.6 and Ch.9

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9.6.26 $${\mathrm{Bi}}^{\prime }\left(z\right)=\frac{3^{1/6}}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}{\mathrm{e}}^{-\zeta }{}_1{}^{}F_1^{}(-\frac{1}{6};-\frac{1}{3};2\zeta )+\frac{3^{7/6}}{2^{7/3}\mathrm{\Gamma }\left(\frac{2}{3}\right)}{\zeta }^{4/3}{\mathrm{e}}^{-\zeta }{}_1{}^{}F_1^{}(\frac{7}{6};\frac{7}{3};2\zeta ).$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $\zeta$: change of variable

Proof sketch:

Combine (9.6.24) with (13.14.2) and refer to §13.1.

Referenced by:

Erratum (V1.0.9) for Equation (9.6.26)

Permalink:

http://dlmf.nist.gov/9.6.E26

Encodings:

pMML, png, TeX

Errata (effective with 1.0.9):

Originally the second occurrence of the function ${}_1{}^{}F_1^{}$ was given incorrectly as ${}_1{}^{}F_1^{}(\frac{7}{6};\frac{7}{3};\zeta )$.

Reported 2014-05-21 by Hanyou Chu

See also:

Annotations for §9.6(iii), §9.6 and Ch.9

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34: 7.18 Repeated Integrals of the Complementary Error Function

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7.18.9 $${\mathrm{i}}^n\mathrm{erfc}\left(z\right)={\mathrm{e}}^{-z^2}\left(\frac{1}{2^n\mathrm{\Gamma }\left(\frac{1}{2}n+1\right)}M(\frac{1}{2}n+\frac{1}{2},\frac{1}{2},z^2)-\frac{z}{2^{n-1}\mathrm{\Gamma }\left(\frac{1}{2}n+\frac{1}{2}\right)}M(\frac{1}{2}n+1,\frac{3}{2},z^2)\right),$$

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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, ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.2.12

Permalink:

http://dlmf.nist.gov/7.18.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(iv), §7.18(iv), §7.18 and Ch.7

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7.18.10 $${\mathrm{i}}^n\mathrm{erfc}\left(z\right)=\frac{{\mathrm{e}}^{-z^2}}{2^n\sqrt{\pi }}U(\frac{1}{2}n+\frac{1}{2},\frac{1}{2},z^2).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§7.18(iv), §7.18(iv), Erratum (V1.0.21) for Paragraph Confluent Hypergeometric Functions (in §7.18(iv))

Permalink:

http://dlmf.nist.gov/7.18.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(iv), §7.18(iv), §7.18 and Ch.7

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35: Bibliography Z

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Distribution of zeros of Gauss and Kummer hypergeometric functions. A semiclassical approach. Ann. Numer. Math. 2 (1-4), pp. 457–472.

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External Links:

ISSN 1021-2655, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§13.9(i), §15.13.

Encodings:

BibTeX

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36: Bibliography T

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Asymptotic expansions of Kummer hypergeometric functions with three asymptotic parameters a, b and z. Indagationes Mathematicae.

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External Links:

ISSN 0019-3577, Document, Link

Referenced by:

§13.8(iv).

Encodings:

BibTeX

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Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters. Integral Transforms Spec. Funct. 33 (1), pp. 16–31.

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External Links:

ISSN 1065-2469, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§13.8(iv).

Encodings:

BibTeX

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37: 16.4 Argument Unity

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§16.4(iii) Identities

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38: 33.14 Definitions and Basic Properties

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33.14.5 $$f(ϵ,\mathrm{\ell };r)={(2r)}^{\mathrm{\ell }+1}{\mathrm{e}}^{-r/\kappa }M(\mathrm{\ell }+1-\kappa ,2\mathrm{\ell }+2,2r/\kappa )/(2\mathrm{\ell }+1)!,$$

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Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter and $\kappa$: quantity

Referenced by:

§13.6(vii)

Permalink:

http://dlmf.nist.gov/33.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §33.14(ii), §33.14 and Ch.33

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39: Bibliography

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Numerical evaluation of the Kummer function with complex argument by the trapezoidal rule. Rend. Sem. Mat. Univ. Politec. Torino 49 (3), pp. 315–327.

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External Links:

ISSN 0373-1243, MathReview, ZentralBlatt

Referenced by:

§13.29(iii).

Encodings:

BibTeX

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40: Bibliography D

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Computational properties of three-term recurrence relations for Kummer functions. J. Comput. Appl. Math. 233 (6), pp. 1505–1510.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§13.29(iv), §13.29(iv).

Encodings:

BibTeX

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