8 Incomplete Gamma and Related Functions Incomplete Gamma Functions 8.1 Special Notation 8.3 Graphics

§8.2 Definitions and Basic Properties

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Keywords:: basic properties, incomplete gamma functions
Permalink:: http://dlmf.nist.gov/8.2
See also:: Annotations for Ch.8

§8.2(i) Definitions

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Keywords:: definitions, general values, incomplete gamma functions, normalized, principal values
Notes:: See Olver (1997b, p. 45) and Temme (1996a).
Referenced by:: §10.17(v), §13.18(ii), §13.6(ii), §13.7(iii), §6.11, §7.11, §7.16, §8.18(ii), §8.21(i), §8.21(ii), §8.8, §9.10(v), §9.7(v)
Permalink:: http://dlmf.nist.gov/8.2.i
See also:: Annotations for §8.2 and Ch.8

The general values of the incomplete gamma functions $\gamma\left(a,z\right)$ and $\Gamma\left(a,z\right)$ are defined by

8.2.1	$\displaystyle\gamma\left(a,z\right)$	$\displaystyle=\int_{0}^{z}t^{a-1}e^{-t}\,\mathrm{d}t,$
$\Re a>0$ ,
ⓘ Defines: $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part, $z$ : complex variable and $a$ : parameter Referenced by: §8.2(iii), §8.6(i), §8.6(ii) Permalink: http://dlmf.nist.gov/8.2.E1 Encodings: TeX, pMML, png See also: Annotations for §8.2(i), §8.2 and Ch.8
8.2.2	$\displaystyle\Gamma\left(a,z\right)$	$\displaystyle=\int_{z}^{\infty}t^{a-1}e^{-t}\,\mathrm{d}t,$
ⓘ Defines: $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $z$ : complex variable and $a$ : parameter A&S Ref: 6.5.3 Referenced by: (8.15.2), §8.2(i), §8.2(iii), §8.21(iii), §8.6(i) Permalink: http://dlmf.nist.gov/8.2.E2 Encodings: TeX, pMML, png See also: Annotations for §8.2(i), §8.2 and Ch.8

without restrictions on the integration paths. However, when the integration paths do not cross the negative real axis, and in the case of (8.2.2) exclude the origen, $\gamma\left(a,z\right)$ and $\Gamma\left(a,z\right)$ take their principal values; compare §4.2(i). Except where indicated otherwise in the DLMF these principal values are assumed. For example,

8.2.3		$\gamma\left(a,z\right)+\Gamma\left(a,z\right)=\Gamma\left(a\right),$
$a\neq 0,-1,-2,\dots$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable and $a$ : parameter A&S Ref: 6.5.3 Referenced by: §8.6(i), §8.7 Permalink: http://dlmf.nist.gov/8.2.E3 Encodings: TeX, pMML, png See also: Annotations for §8.2(i), §8.2 and Ch.8

Normalized functions are:

8.2.4	$\displaystyle P\left(a,z\right)$	$\displaystyle=\frac{\gamma\left(a,z\right)}{\Gamma\left(a\right)},$
8.2.4	$\displaystyle Q\left(a,z\right)$	$\displaystyle=\frac{\Gamma\left(a,z\right)}{\Gamma\left(a\right)},$
ⓘ Defines: $P\left(\NVar{a},\NVar{z}\right)$ : normalized incomplete gamma function and $Q\left(\NVar{a},\NVar{z}\right)$ : normalized incomplete gamma function Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable and $a$ : parameter A&S Ref: 6.5.1 (The function $Q\left(a,z\right)$ is not defined in AMS 55.) Permalink: http://dlmf.nist.gov/8.2.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §8.2(i), §8.2 and Ch.8

8.2.5		$P\left(a,z\right)+Q\left(a,z\right)=1.$
ⓘ Symbols: $P\left(\NVar{a},\NVar{z}\right)$ : normalized incomplete gamma function, $Q\left(\NVar{a},\NVar{z}\right)$ : normalized incomplete gamma function, $z$ : complex variable and $a$ : parameter Permalink: http://dlmf.nist.gov/8.2.E5 Encodings: TeX, pMML, png See also: Annotations for §8.2(i), §8.2 and Ch.8

In addition,

8.2.6		$\gamma^{*}\left(a,z\right)=z^{-a}P\left(a,z\right)=\frac{z^{-a}}{\Gamma\left(a% \right)}\gamma\left(a,z\right).$
ⓘ Defines: $\gamma^{*}\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $P\left(\NVar{a},\NVar{z}\right)$ : normalized incomplete gamma function, $z$ : complex variable and $a$ : parameter A&S Ref: 6.5.4 Referenced by: §8.2(iii), §8.6(i) Permalink: http://dlmf.nist.gov/8.2.E6 Encodings: TeX, pMML, png See also: Annotations for §8.2(i), §8.2 and Ch.8

8.2.7		$\gamma^{*}\left(a,z\right)=\frac{1}{\Gamma\left(a\right)}\int_{0}^{1}t^{a-1}e^% {-zt}\,\mathrm{d}t,$
$\Re a>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\gamma^{*}\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral, $\Re$ : real part, $z$ : complex variable and $a$ : parameter Permalink: http://dlmf.nist.gov/8.2.E7 Encodings: TeX, pMML, png See also: Annotations for §8.2(i), §8.2 and Ch.8

§8.2(ii) Analytic Continuation

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Keywords:: analytic continuation, incomplete gamma functions
Notes:: See Olver (1997b, p. 45) and Temme (1996a).
Referenced by:: §8.21(i)
Permalink:: http://dlmf.nist.gov/8.2.ii
See also:: Annotations for §8.2 and Ch.8

In this subsection the functions $\gamma$ and $\Gamma$ have their general values.

The function $\gamma^{*}\left(a,z\right)$ is entire in $z$ and $a$ . When $z\neq 0$ , $\Gamma\left(a,z\right)$ is an entire function of $a$ , and $\gamma\left(a,z\right)$ is meromorphic with simple poles at $a=-n$ , $n=0,1,2,\dots$ , with residue $(-1)^{n}/n!$ .

For $m\in\mathbb{Z}$ ,

8.2.8	$\displaystyle\gamma\left(a,ze^{2\pi mi}\right)$	$\displaystyle=e^{2\pi mia}\gamma\left(a,z\right),$
$a\neq 0,-1,-2,\dots$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable, $a$ : parameter and $m$ : integer Permalink: http://dlmf.nist.gov/8.2.E8 Encodings: TeX, pMML, png See also: Annotations for §8.2(ii), §8.2 and Ch.8
8.2.9	$\displaystyle\Gamma\left(a,ze^{2\pi mi}\right)$	$\displaystyle=e^{2\pi mia}\Gamma\left(a,z\right)+(1-e^{2\pi mia})\Gamma\left(a% \right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable, $a$ : parameter and $m$ : integer Referenced by: §8.2(ii) Permalink: http://dlmf.nist.gov/8.2.E9 Encodings: TeX, pMML, png See also: Annotations for §8.2(ii), §8.2 and Ch.8

(8.2.9) also holds when $a$ is zero or a negative integer, provided that the right-hand side is replaced by its limiting value. For example, in the case $m=-1$ we have

8.2.10		$e^{-\pi ia}\Gamma\left(a,ze^{\pi i}\right)-e^{\pi ia}\Gamma\left(a,ze^{-\pi i}% \right)=-\frac{2\pi i}{\Gamma\left(1-a\right)},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable and $a$ : parameter Permalink: http://dlmf.nist.gov/8.2.E10 Encodings: TeX, pMML, png See also: Annotations for §8.2(ii), §8.2 and Ch.8

without restriction on $a$ .

Lastly,

8.2.11		$\Gamma\left(a,ze^{\pm\pi i}\right)=\Gamma\left(a\right)(1-z^{a}e^{\pm\pi ia}% \gamma^{*}\left(a,-z\right)).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\gamma^{*}\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable and $a$ : parameter Permalink: http://dlmf.nist.gov/8.2.E11 Encodings: TeX, pMML, png See also: Annotations for §8.2(ii), §8.2 and Ch.8

§8.2(iii) Differential Equations

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Keywords:: differential equations, incomplete gamma functions
Notes:: Use (8.2.1), (8.2.2), and (8.2.6).
Permalink:: http://dlmf.nist.gov/8.2.iii
See also:: Annotations for §8.2 and Ch.8

If $w=\gamma\left(a,z\right)$ or $\Gamma\left(a,z\right)$ , then

8.2.12		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(1+\frac{1-a}{z}\right)\frac{% \mathrm{d}w}{\mathrm{d}z}=0.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $a$ : parameter and $w$ : solution Permalink: http://dlmf.nist.gov/8.2.E12 Encodings: TeX, pMML, png See also: Annotations for §8.2(iii), §8.2 and Ch.8

If $w=e^{z}z^{1-a}\Gamma\left(a,z\right)$ , then

8.2.13		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-\left(1+\frac{1-a}{z}\right)\frac{% \mathrm{d}w}{\mathrm{d}z}+\frac{1-a}{z^{2}}w=0.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $a$ : parameter and $w$ : solution Permalink: http://dlmf.nist.gov/8.2.E13 Encodings: TeX, pMML, png See also: Annotations for §8.2(iii), §8.2 and Ch.8

Also,

8.2.14		$z\frac{{\mathrm{d}}^{2}\gamma^{}}{{\mathrm{d}z}^{2}}+(a+1+z)\frac{\mathrm{d}% \gamma^{}}{\mathrm{d}z}+a\gamma^{*}=0.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\gamma^{*}\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable and $a$ : parameter A&S Ref: 6.5.28 Permalink: http://dlmf.nist.gov/8.2.E14 Encodings: TeX, pMML, png See also: Annotations for §8.2(iii), §8.2 and Ch.8

8.1 Special Notation 8.3 Graphics

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§8.2 Definitions and Basic Properties

Contents

§8.2(i) Definitions

§8.2(ii) Analytic Continuation

§8.2(iii) Differential Equations

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