8 Incomplete Gamma and Related Functions Incomplete Gamma Functions 8.3 Graphics 8.5 Confluent Hypergeometric Representations

§8.4 Special Values

Notes:: See Erdélyi et al. (1953b, Chapter 9). (8.4.14) follows from (8.4.6) and (8.8.6). (8.4.15) follows from the first series in (8.7.3) by combining $\mathop{\Gamma\/}\nolimits\!\left(a\right)$ with the term of the series, then taking the limit as $a\rightarrow-n$ .
Keywords:: incomplete gamma functions
Permalink:: http://dlmf.nist.gov/8.4

For $\mathop{\mathrm{erf}\/}\nolimits\!\left(z\right)$ , $\mathop{\mathrm{erfc}\/}\nolimits\!\left(z\right)$ , and $\mathop{F\/}\nolimits\!\left(z\right)$ , see §§7.2(i), 7.2(ii). For $\mathop{E_{{n}}\/}\nolimits\!\left(z\right)$ see §8.19(i).

8.4.1 $\mathop{\gamma\/}\nolimits\!\left(\tfrac{1}{2},z^{2}\right)=2\int_{0}^{z}e^{{-% t^{2}}}dt=\sqrt{\pi}\mathop{\mathrm{erf}\/}\nolimits\!\left(z\right),$

Symbols:: : differential of , $\mathop{\mathrm{erf}\/}\nolimits z$ : error function, : base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral and : complex variable
A&S Ref:: 6.5.16
Permalink:: http://dlmf.nist.gov/8.4.E1
Encodings:: TeX, pMML, png

8.4.2 $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,0\right)=\frac{1}{\mathop{\Gamma\/}% \nolimits\!\left(a+1\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function and : parameter
Permalink:: http://dlmf.nist.gov/8.4.E2
Encodings:: TeX, pMML, png

8.4.3 $\mathop{\gamma^{{*}}\/}\nolimits\!\left(\tfrac{1}{2},-z^{2}\right)=\frac{2e^{{% z^{2}}}}{z\sqrt{\pi}}\mathop{F\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(z\right)$ : Dawson’s integral, : base of exponential function, $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function and : complex variable
A&S Ref:: 6.5.18
Permalink:: http://dlmf.nist.gov/8.4.E3
Encodings:: TeX, pMML, png

8.4.4 $\mathop{\Gamma\/}\nolimits\!\left(0,z\right)=\int_{z}^{\infty}t^{{-1}}e^{{-t}}% dt=\mathop{E_{1}\/}\nolimits\!\left(z\right),$

Symbols:: : differential of , : base of exponential function, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral and : complex variable
A&S Ref:: 6.5.15
Permalink:: http://dlmf.nist.gov/8.4.E4
Encodings:: TeX, pMML, png

8.4.5 $\mathop{\Gamma\/}\nolimits\!\left(1,z\right)=e^{{-z}},$

Symbols:: : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function and : complex variable
Permalink:: http://dlmf.nist.gov/8.4.E5
Encodings:: TeX, pMML, png

8.4.6 $\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2},z^{2}\right)=2\int_{z}^{\infty}% e^{{-t^{2}}}dt=\sqrt{\pi}\mathop{\mathrm{erfc}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, : differential of , : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral and : complex variable
A&S Ref:: 6.5.17
Referenced by:: §8.4
Permalink:: http://dlmf.nist.gov/8.4.E6
Encodings:: TeX, pMML, png

For $n=0,1,2,\dots$ ,

8.4.7 $\mathop{\gamma\/}\nolimits\!\left(n+1,z\right)=n!(1-e^{{-z}}e_{n}(z)),$

Symbols:: : base of exponential function, : : factorial, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, : complex variable, : nonnegative integer and $e_{n}(z)$ : functions
Referenced by:: §8.11(v)
Permalink:: http://dlmf.nist.gov/8.4.E7
Encodings:: TeX, pMML, png

8.4.8 $\mathop{\Gamma\/}\nolimits\!\left(n+1,z\right)=n!e^{{-z}}e_{n}(z),$

Symbols:: : base of exponential function, : : factorial, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, : complex variable, : nonnegative integer and $e_{n}(z)$ : functions
Permalink:: http://dlmf.nist.gov/8.4.E8
Encodings:: TeX, pMML, png

8.4.9 $\mathop{P\/}\nolimits\!\left(n+1,z\right)=1-e^{{-z}}e_{n}(z),$

Symbols:: : base of exponential function, $\mathop{P\/}\nolimits\!\left(a,z\right)$ : normalized incomplete gamma function, : complex variable, : nonnegative integer and $e_{n}(z)$ : functions
A&S Ref:: 6.5.13
Permalink:: http://dlmf.nist.gov/8.4.E9
Encodings:: TeX, pMML, png

8.4.10 $\mathop{Q\/}\nolimits\!\left(n+1,z\right)=e^{{-z}}e_{n}(z),$

Symbols:: : base of exponential function, $\mathop{Q\/}\nolimits\!\left(a,z\right)$ : normalized incomplete gamma function, : complex variable, : nonnegative integer and $e_{n}(z)$ : functions
Permalink:: http://dlmf.nist.gov/8.4.E10
Encodings:: TeX, pMML, png

where

8.4.11 $e_{n}(z)=\sum_{{k=0}}^{n}\frac{z^{k}}{k!}.$

Symbols:: : : factorial, : complex variable, : nonnegative integer, : nonnegative integer and $e_{n}(z)$ : functions
A&S Ref:: 6.5.11
Referenced by:: §8.11(v), §8.7
Permalink:: http://dlmf.nist.gov/8.4.E11
Encodings:: TeX, pMML, png

Also

8.4.12 $\mathop{\gamma^{{*}}\/}\nolimits\!\left(-n,z\right)=z^{n},$

Symbols:: $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, : complex variable and : nonnegative integer
A&S Ref:: 6.5.14
Referenced by:: §8.13(i)
Permalink:: http://dlmf.nist.gov/8.4.E12
Encodings:: TeX, pMML, png

8.4.13 $\mathop{\Gamma\/}\nolimits\!\left(1-n,z\right)=z^{{1-n}}\mathop{E_{{n}}\/}% \nolimits\!\left(z\right),$

Symbols:: $\mathop{E_{{p}}\/}\nolimits\!\left(z\right)$ : generalized exponential integral, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, : complex variable and : nonnegative integer
Referenced by:: §8.19(iv)
Permalink:: http://dlmf.nist.gov/8.4.E13
Encodings:: TeX, pMML, png

8.4.14 $\mathop{Q\/}\nolimits\!\left(n+\tfrac{1}{2},z^{2}\right)=\mathop{\mathrm{erfc}% \/}\nolimits\!\left(z\right)+\frac{e^{{-z^{2}}}}{\sqrt{\pi}}\sum_{{k=1}}^{n}% \frac{z^{{2k-1}}}{\left(\tfrac{1}{2}\right)_{{k}}},$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, : base of exponential function, $\mathop{Q\/}\nolimits\!\left(a,z\right)$ : normalized incomplete gamma function, : complex variable, : nonnegative integer and : nonnegative integer
Referenced by:: §8.4
Permalink:: http://dlmf.nist.gov/8.4.E14
Encodings:: TeX, pMML, png

8.4.15 $\mathop{\Gamma\/}\nolimits\!\left(-n,z\right)=\frac{(-1)^{n}}{n!}\left(\mathop% {E_{1}\/}\nolimits\!\left(z\right)-e^{{-z}}\sum_{{k=0}}^{{n-1}}\frac{(-1)^{k}k% !}{z^{{k+1}}}\right)=\frac{(-1)^{n}}{n!}\left(\mathop{\psi\/}\nolimits\!\left(% n+1\right)-\mathop{\ln\/}\nolimits z\right)-z^{{-n}}\sum_{{\substack{k=0\\ k\neq n}}}^{{\infty}}\frac{(-z)^{k}}{k!(k-n)}.$

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, : base of exponential function, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, : : factorial, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : complex variable, : nonnegative integer and : nonnegative integer
A&S Ref:: 6.5.19
Referenced by:: §8.19(iii), §8.19(iv), §8.4
Permalink:: http://dlmf.nist.gov/8.4.E15
Encodings:: TeX, pMML, png

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