# §8.2 Definitions and Basic Properties

## §8.2(i) Definitions

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, $\mathrm{e}$: 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, $\mathrm{e}$: 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 origin, $\gamma\left(a,z\right)$ 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 $\displaystyle P\left(a,z\right)$ $\displaystyle=\frac{\gamma\left(a,z\right)}{\Gamma\left(a\right)},$ $\displaystyle Q\left(a,z\right)$ $\displaystyle=\frac{\Gamma\left(a,z\right)}{\Gamma\left(a\right)},$ ⓘ 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 $P\left(a,z\right)+Q\left(a,z\right)=1.$

 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 $\gamma^{*}\left(a,z\right)=\frac{1}{\Gamma\left(a\right)}\int_{0}^{1}t^{a-1}e^% {-zt}\mathrm{d}t,$ $\Re a>0$.

## §8.2(ii) Analytic Continuation

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 $\displaystyle\gamma\left(a,ze^{2\pi mi}\right)$ $\displaystyle=e^{2\pi mia}\gamma\left(a,z\right),$ $a\neq 0,-1,-2,\dots$, 8.2.9 $\displaystyle\Gamma\left(a,ze^{2\pi mi}\right)$ $\displaystyle=e^{2\pi mia}\Gamma\left(a,z\right)+(1-e^{2\pi mia})\Gamma\left(a% \right).$

(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 $e^{-\pi ia}\Gamma\left(a,ze^{\pi i}\right)-e^{\pi ia}\Gamma\left(a,ze^{-\pi i}% \right)=-\frac{2\pi i}{\Gamma\left(1-a\right)},$

without restriction on $a$.

Lastly,

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

## §8.2(iii) Differential Equations

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, $z$: 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, $z$: 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, $\gamma^{*}\left(\NVar{a},\NVar{z}\right)$: 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