8 Incomplete Gamma and Related FunctionsIncomplete Gamma Functions8.4 Special Values8.6 Integral Representations

§8.5 Confluent Hypergeometric Representations

ⓘ

Keywords:

confluent hypergeometric functions, incomplete gamma functions, relations to other functions

Notes:

See Slater (1960, §5.6) and §13.2(vii). For (8.5.4) see (13.18.4).

Permalink:

http://dlmf.nist.gov/8.5

See also:

Annotations for Ch.8

For the confluent hypergeometric functions $M$, $𝐌$, $U$, and the Whittaker functions $M_{\kappa ,\mu }$ and $W_{\kappa ,\mu }$, see §§13.2(i) and 13.14(i).

8.5.1		$$\gamma (a,z)=a^{-1}{z}^a{\mathrm{e}}^{-z}M(1,1+a,z)=a^{-1}{z}^{a}M(a,1+a,-z),$$
$a\ne 0,-1,-2,\mathrm{\dots }$.
ⓘ Symbols: $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter A&S Ref: 6.5.12 Referenced by: §8.11(ii), §8.14 Permalink: http://dlmf.nist.gov/8.5.E1 Encodings: TeX, pMML, png See also: Annotations for §8.5 and Ch.8

8.5.2		$${\gamma }^{\ast }(a,z)={\mathrm{e}}^{-z}𝐌(1,1+a,z)=𝐌(a,1+a,-z).$$
ⓘ Symbols: $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter Referenced by: §8.7 Permalink: http://dlmf.nist.gov/8.5.E2 Encodings: TeX, pMML, png See also: Annotations for §8.5 and Ch.8

8.5.3		$$\mathrm{\Gamma }(a,z)={\mathrm{e}}^{-z}U(1-a,1-a,z)=z^a{\mathrm{e}}^{-z}U(1,1+a,z).$$
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter Referenced by: §8.14, §8.19(vi) Permalink: http://dlmf.nist.gov/8.5.E3 Encodings: TeX, pMML, png See also: Annotations for §8.5 and Ch.8

8.5.4		$$\gamma (a,z)=a^{-1}{z}^{\frac{1}{2}a-\frac{1}{2}}{\mathrm{e}}^{-\frac{1}{2}z}{M}_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$$
ⓘ Symbols: $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter Referenced by: §8.5 Permalink: http://dlmf.nist.gov/8.5.E4 Encodings: TeX, pMML, png See also: Annotations for §8.5 and Ch.8

8.5.5		$$\mathrm{\Gamma }(a,z)={\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}a-\frac{1}{2}}{W}_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$$
ⓘ Symbols: $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter Permalink: http://dlmf.nist.gov/8.5.E5 Encodings: TeX, pMML, png See also: Annotations for §8.5 and Ch.8

8.4 Special Values8.6 Integral Representations

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