§8.2 Definitions and Basic Properties

Keywords:: incomplete gamma functions
Permalink:: http://dlmf.nist.gov/8.2

§8.2(i) Definitions
§8.2(ii) Analytic Continuation
§8.2(iii) Differential Equations

§8.2(i) Definitions

Notes:: See Olver (1997b, p. 45) and Temme (1996a).
Keywords:: incomplete gamma functions, principal values
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

The general values of the incomplete gamma functions $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ and $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ are defined by

8.2.1 $\mathop{\gamma\/}\nolimits\!\left(a,z\right)=\int _{0}^{z}t^{{a-1}}e^{{-t}}dt,$ $\realpart{a}>0$ ,

Defines:: $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function
Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : 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
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8.2.2 $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)=\int _{z}^{\infty}t^{{a-1}}e^{{-t}}dt,$

Defines:: $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function
Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $z$ : complex variable and $a$ : parameter
A&S Ref:: 6.5.3
Referenced by:: §8.2(i), §8.2(iii), §8.21(iii), §8.6(i)
Permalink:: http://dlmf.nist.gov/8.2.E2
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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 origin, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ and $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ 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 $\mathop{\gamma\/}\nolimits\!\left(a,z\right)+\mathop{\Gamma\/}\nolimits\!\left(a,z\right)=\mathop{\Gamma\/}\nolimits\!\left(a\right),$ $a\neq 0,-1,-2,\dots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : 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
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Normalized functions are:

8.2.4

$\mathop{P\/}\nolimits\!\left(a,z\right)=\frac{\mathop{\gamma\/}\nolimits\!\left(a,z\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)},$

$\mathop{Q\/}\nolimits\!\left(a,z\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(a,z\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)},$

Defines:: $\mathop{P\/}\nolimits\!\left(a,z\right)$ : normalized incomplete gamma function and $\mathop{Q\/}\nolimits\!\left(a,z\right)$ : normalized incomplete gamma function
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $z$ : complex variable and $a$ : parameter
A&S Ref:: 6.5.1 (The function $\mathop{Q\/}\nolimits\!\left(a,z\right)$ is not defined in AMS 55.)
Referenced by:: §8.11(v)
Permalink:: http://dlmf.nist.gov/8.2.E4
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8.2.5 $\mathop{P\/}\nolimits\!\left(a,z\right)+\mathop{Q\/}\nolimits\!\left(a,z\right)=1.$

Symbols:: $\mathop{P\/}\nolimits\!\left(a,z\right)$ : normalized incomplete gamma function, $\mathop{Q\/}\nolimits\!\left(a,z\right)$ : normalized incomplete gamma function, $z$ : complex variable and $a$ : parameter
Permalink:: http://dlmf.nist.gov/8.2.E5
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In addition,

8.2.6 $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)=z^{{-a}}\mathop{P\/}\nolimits\!\left(a,z\right)=\frac{z^{{-a}}}{\mathop{\Gamma\/}\nolimits\!\left(a\right)}\mathop{\gamma\/}\nolimits\!\left(a,z\right).$

Defines:: $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\mathop{P\/}\nolimits\!\left(a,z\right)$ : 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
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8.2.7 $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(a\right)}\int _{0}^{1}t^{{a-1}}e^{{-zt}}dt,$ $\realpart{a}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral, $\realpart{}$ : real part, $z$ : complex variable and $a$ : parameter
Permalink:: http://dlmf.nist.gov/8.2.E7
Encodings:: TeX, pMML, png

§8.2(ii) Analytic Continuation

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

In this subsection the functions $\mathop{\gamma\/}\nolimits$ and $\mathop{\Gamma\/}\nolimits$ have their general values.

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

For $m\in\Integer$ ,

8.2.8 $\mathop{\gamma\/}\nolimits\!\left(a,ze^{{2\pi mi}}\right)=e^{{2\pi mia}}\mathop{\gamma\/}\nolimits\!\left(a,z\right),$ $a\neq 0,-1,-2,\dots$ ,

Symbols:: $e$ : base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $z$ : complex variable, $a$ : parameter and $m$ : integer
Permalink:: http://dlmf.nist.gov/8.2.E8
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8.2.9 $\mathop{\Gamma\/}\nolimits\!\left(a,ze^{{2\pi mi}}\right)=e^{{2\pi mia}}\mathop{\Gamma\/}\nolimits\!\left(a,z\right)+(1-e^{{2\pi mia}})\mathop{\Gamma\/}\nolimits\!\left(a\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $z$ : complex variable, $a$ : parameter and $m$ : integer
Referenced by:: §8.2(ii)
Permalink:: http://dlmf.nist.gov/8.2.E9
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(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 $e^{{-\pi ia}}\mathop{\Gamma\/}\nolimits\!\left(a,ze^{{\pi i}}\right)-e^{{\pi ia}}\mathop{\Gamma\/}\nolimits\!\left(a,ze^{{-\pi i}}\right)=-\frac{2\pi i}{\mathop{\Gamma\/}\nolimits\!\left(1-a\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $z$ : complex variable and $a$ : parameter
Permalink:: http://dlmf.nist.gov/8.2.E10
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without restriction on $a$ .

Lastly,

8.2.11 $\mathop{\Gamma\/}\nolimits\!\left(a,ze^{{\pm\pi i}}\right)=\mathop{\Gamma\/}\nolimits\!\left(a\right)(1-z^{a}e^{{\pm\pi ia}}\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,-z\right)).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $z$ : complex variable and $a$ : parameter
Permalink:: http://dlmf.nist.gov/8.2.E11
Encodings:: TeX, pMML, png

§8.2(iii) Differential Equations

Notes:: Use (8.2.1), (8.2.2), and (8.2.6).
Keywords:: differential equations, incomplete gamma functions
Permalink:: http://dlmf.nist.gov/8.2.iii

If $w=\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ or $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ , then

8.2.12 $\frac{{d}^{2}w}{{dz}^{2}}+\left(1+\frac{1-a}{z}\right)\frac{dw}{dz}=0.$

Symbols:: $\frac{df}{dx}$ : 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

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

8.2.13 $\frac{{d}^{2}w}{{dz}^{2}}-\left(1+\frac{1-a}{z}\right)\frac{dw}{dz}+\frac{1-a}{z^{2}}w=0.$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $a$ : parameter and $w$ : solution
Permalink:: http://dlmf.nist.gov/8.2.E13
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Also,

8.2.14 $z\frac{{d}^{2}\mathop{\gamma^{{*}}\/}\nolimits}{{dz}^{2}}+(a+1+z)\frac{d\mathop{\gamma^{{*}}\/}\nolimits}{dz}+a\mathop{\gamma^{{*}}\/}\nolimits=0.$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $z$ : complex variable and $a$ : parameter
A&S Ref:: 6.5.28
Permalink:: http://dlmf.nist.gov/8.2.E14
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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