# §8.2 Definitions and Basic Properties

## §8.2(i) Definitions

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 $\displaystyle\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ $\displaystyle=\int_{0}^{z}t^{a-1}e^{-t}\mathrm{d}t,$ $\Re{a}>0$, Defines: $\mathop{\gamma\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: incomplete gamma function Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of exponential function, $\int$: 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) 8.2.2 $\displaystyle\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ $\displaystyle=\int_{z}^{\infty}t^{a-1}e^{-t}\mathrm{d}t,$ Defines: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: incomplete gamma function Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: 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 Encodings: TeX, pMML, png See also: Annotations for 8.2(i)

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 $\mathop{\gamma\/}\nolimits\!\left(a,z\right)+\mathop{\Gamma\/}\nolimits\!\left% (a,z\right)=\mathop{\Gamma\/}\nolimits\!\left(a\right),$ $a\neq 0,-1,-2,\dots$.

Normalized functions are:

 8.2.4 $\displaystyle\mathop{P\/}\nolimits\!\left(a,z\right)$ $\displaystyle=\frac{\mathop{\gamma\/}\nolimits\!\left(a,z\right)}{\mathop{% \Gamma\/}\nolimits\!\left(a\right)},$ $\displaystyle\mathop{Q\/}\nolimits\!\left(a,z\right)$ $\displaystyle=\frac{\mathop{\Gamma\/}\nolimits\!\left(a,z\right)}{\mathop{% \Gamma\/}\nolimits\!\left(a\right)},$ Defines: $\mathop{P\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: normalized incomplete gamma function and $\mathop{Q\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: normalized incomplete gamma function Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathop{\Gamma\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: incomplete gamma function, $\mathop{\gamma\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: 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.) Permalink: http://dlmf.nist.gov/8.2.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 8.2(i)
 8.2.5 $\mathop{P\/}\nolimits\!\left(a,z\right)+\mathop{Q\/}\nolimits\!\left(a,z\right% )=1.$

 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: $\mathop{\gamma^{*}\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: incomplete gamma function Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathop{\gamma\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: incomplete gamma function, $\mathop{P\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: 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)
 8.2.7 $\mathop{\gamma^{*}\/}\nolimits\!\left(a,z\right)=\frac{1}{\mathop{\Gamma\/}% \nolimits\!\left(a\right)}\int_{0}^{1}t^{a-1}e^{-zt}\mathrm{d}t,$ $\Re{a}>0$.

## §8.2(ii) Analytic Continuation

In this subsection the functions $\mathop{\gamma\/}\nolimits$ 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 $m\in\mathbb{Z}$,

 8.2.8 $\displaystyle\mathop{\gamma\/}\nolimits\!\left(a,ze^{2\pi mi}\right)$ $\displaystyle=e^{2\pi mia}\mathop{\gamma\/}\nolimits\!\left(a,z\right),$ $a\neq 0,-1,-2,\dots$, 8.2.9 $\displaystyle\mathop{\Gamma\/}\nolimits\!\left(a,ze^{2\pi mi}\right)$ $\displaystyle=e^{2\pi mia}\mathop{\Gamma\/}\nolimits\!\left(a,z\right)+(1-e^{2% \pi mia})\mathop{\Gamma\/}\nolimits\!\left(a\right).$

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

without restriction on $a$.

Lastly,

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

## §8.2(iii) Differential Equations

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

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

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

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

Also,

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