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8 Incomplete Gamma and Related Functions Incomplete Gamma Functions 8.4 Special Values 8.6 Integral Representations

§8.5 Confluent Hypergeometric Representations

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

8.5.1		$\gamma\left(a,z\right)=a^{-1}z^{a}e^{-z}M\left(1,1+a,z\right)=a^{-1}z^{a}M% \left(a,1+a,-z\right),$
$a\neq 0,-1,-2,\dots$ .
ⓘ Symbols: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $\gamma\left(\NVar{a},\NVar{z}\right)$ : 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		$\gamma^{*}\left(a,z\right)=e^{-z}{\mathbf{M}}\left(1,1+a,z\right)={\mathbf{M}}% \left(a,1+a,-z\right).$
ⓘ Symbols: ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $\gamma^{*}\left(\NVar{a},\NVar{z}\right)$ : 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		$\Gamma\left(a,z\right)=e^{-z}U\left(1-a,1-a,z\right)=z^{a}e^{-z}U\left(1,1+a,z% \right).$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : 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		$\gamma\left(a,z\right)=a^{-1}z^{\frac{1}{2}a-\frac{1}{2}}e^{-\frac{1}{2}z}M_{% \frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$
ⓘ Symbols: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $\gamma\left(\NVar{a},\NVar{z}\right)$ : 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		$\Gamma\left(a,z\right)=e^{-\frac{1}{2}z}z^{\frac{1}{2}a-\frac{1}{2}}W_{\frac{1% }{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : 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 Values 8.6 Integral Representations

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