incomplete gamma functions

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21: 8.13 Zeros

§8.13 Zeros

… ►The function ${\gamma }^{\ast }(a,x)$ has no real zeros for $a\ge 0$. … ►When $-5\le a\le 4$ the behavior of the $x$-zeros as functions of $a$ can be seen by taking the slice ${\gamma }^{\ast }(a,x)=0$ of the surface depicted in Figure 8.3.6. … ►

Table 8.13.1: Double zeros $(a_n^{\ast },x_n^{\ast })$ of ${\gamma }^{\ast }(a,x)$.

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$n$	$a_n^{\ast }$		$x_n^{\ast }$
…

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22: 8.6 Integral Representations

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§8.6(i) Integrals Along the Real Line

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§8.6(ii) Contour Integrals

… ► … ►

Mellin–Barnes Integrals

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§8.6(iii) Compendia

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23: 8.15 Sums

§8.15 Sums

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8.15.1 $$\gamma (a,\lambda x)={\lambda }^a\sum _{k=0}^{\mathrm{\infty }}\gamma (a+k,x)\frac{{(1-\lambda )}^k}{k!}.$$

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Symbols:

$!$: factorial (as in $n!$), $\gamma (a,z)$: incomplete gamma function, $x$: real variable, $a$: parameter and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/8.15.E1

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Annotations for §8.15 and Ch.8

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8.15.2 $$a\sum _{k=1}^{\mathrm{\infty }}\left(\frac{{\mathrm{e}}^{2\pi \mathrm{i}k(z+h)}}{{\left(2\pi \mathrm{i}k\right)}^{a+1}}\mathrm{\Gamma }(a,2\pi \mathrm{i}kz)+\frac{{\mathrm{e}}^{-2\pi \mathrm{i}k(z+h)}}{{\left(-2\pi \mathrm{i}k\right)}^{a+1}}\mathrm{\Gamma }(a,-2\pi \mathrm{i}kz)\right)=\zeta (-a,z+h)+\frac{z^{a+1}}{a+1}+\left(h-\frac{1}{2}\right)z^a,$$ $h\in [0,1]$.

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Proof sketch:

Start with and $z>0$. For $\mathrm{\Gamma }(a,\pm 2\pi \mathrm{i}kz)$ take integral representation (8.2.2) and use the substitution $t=\pm 2\pi \mathrm{i}kz(1+\tau )$. The sum and the integral can be interchanged, and the sum can be evaluated via (4.6.1). Use integration by parts. This will result in $\left(h-\frac{1}{2}\right)z^a$ plus two integrals with infinitely many poles. The residue theorem (§1.10(iv)) will give us an infinite series which can be identified via (25.11.1). For other values of $a$ and $z$ use analytic continuation.

Referenced by:

§25.11(x), §25.11(x), §8.15, Erratum (V1.1.3) for Additions

Permalink:

http://dlmf.nist.gov/8.15.E2

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Addition (effective with 1.1.3):

This equation was added.

See also:

Annotations for §8.15 and Ch.8

… ►For other infinite series whose terms include incomplete gamma functions, see Nemes (2017a), Reynolds and Stauffer (2021), and Prudnikov et al. (1986b, §5.2).

24: 8.14 Integrals

§8.14 Integrals

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8.14.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-ax}\frac{\gamma (b,x)}{\mathrm{\Gamma }\left(b\right)}dx=\frac{{(1+a)}^{-b}}{a},$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }b>-1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $a$: parameter

Referenced by:

§8.14, §8.14

Permalink:

http://dlmf.nist.gov/8.14.E1

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Annotations for §8.14 and Ch.8

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8.14.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-ax}\mathrm{\Gamma }(b,x)dx=\mathrm{\Gamma }\left(b\right)\frac{1-{(1+a)}^{-b}}{a},$$ $\mathrm{\Re }a>-1$, $\mathrm{\Re }b>-1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $a$: parameter

A&S Ref:

6.5.36

Referenced by:

§8.14, §8.14

Permalink:

http://dlmf.nist.gov/8.14.E2

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Annotations for §8.14 and Ch.8

… ►

8.14.3 $${\int }_0^{\mathrm{\infty }}{x}^{a-1}\gamma (b,x)dx=-\frac{\mathrm{\Gamma }\left(a+b\right)}{a},$$ , $\mathrm{\Re }\left(a+b\right)>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\gamma (a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $a$: parameter

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/8.14.E3

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Annotations for §8.14 and Ch.8

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8.14.4 $${\int }_0^{\mathrm{\infty }}{x}^{a-1}\mathrm{\Gamma }(b,x)dx=\frac{\mathrm{\Gamma }\left(a+b\right)}{a},$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }\left(a+b\right)>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $a$: parameter

A&S Ref:

6.5.37

Permalink:

http://dlmf.nist.gov/8.14.E4

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Annotations for §8.14 and Ch.8

…

25: 13.18 Relations to Other Functions

… ►

§13.18(ii) Incomplete Gamma Functions

… ►When $\frac{1}{2}-\kappa \pm \mu$ is an integer the Whittaker functions can be expressed as incomplete gamma functions (or generalized exponential integrals). … ►

13.18.4 $$M_{\mu -\frac{1}{2},\mu }\left(z\right)=2\mu {\mathrm{e}}^{\frac{1}{2}z}{z}^{\frac{1}{2}-\mu }\gamma (2\mu ,z),$$

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Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function and $z$: complex variable

Referenced by:

§8.5

Permalink:

http://dlmf.nist.gov/13.18.E4

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See also:

Annotations for §13.18(ii), §13.18 and Ch.13

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13.18.5 $$W_{\mu -\frac{1}{2},\mu }\left(z\right)={\mathrm{e}}^{\frac{1}{2}z}{z}^{\frac{1}{2}-\mu }\mathrm{\Gamma }(2\mu ,z).$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.18.E5

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See also:

Annotations for §13.18(ii), §13.18 and Ch.13

…

26: Bibliography T

… ►

The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. Anal. 10 (4), pp. 757–766.

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External Links:

ISSN 0036-1410, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§8.12.

Encodings:

BibTeX

… ►

Asymptotic inversion of incomplete gamma functions. Math. Comp. 58 (198), pp. 755–764.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§8.12, §8.12.

Encodings:

BibTeX

… ►

A set of algorithms for the incomplete gamma functions. Probab. Engrg. Inform. Sci. 8, pp. 291–307.

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Notes:

Includes Pascal program

External Links:

FullText

Referenced by:

§5.24(ii), §7.25(ii), §8.28(ii).

Encodings:

BibTeX

… ►

Asymptotics of zeros of incomplete gamma functions. Ann. Numer. Math. 2 (1-4), pp. 415–423.

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Notes:

Special functions (Torino, 1993)

External Links:

ISSN 1021-2655, FullText, MathReview (E. R. Love), ZentralBlatt

Referenced by:

§8.13(ii).

Encodings:

BibTeX

… ►

Algorithm 926: incomplete gamma functions with negative arguments. ACM Trans. Math. Software 39 (2), pp. Art. 14, 9.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

7th item, §8.28(ii).

Encodings:

BibTeX

…

27: 8.11 Asymptotic Approximations and Expansions

§8.11 Asymptotic Approximations and Expansions

… ►where $\delta$ denotes an arbitrary small positive constant. … ►

8.11.4 $$\gamma (a,z)=z^a{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{z^k}{{\left(a\right)}_{k+1}},$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

A&S Ref:

6.5.33 (corrected)

Referenced by:

§8.11(ii)

Permalink:

http://dlmf.nist.gov/8.11.E4

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Annotations for §8.11(ii), §8.11 and Ch.8

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8.11.15 $$S_n(x)=\frac{\gamma (n+1,nx)}{{(nx)}^n{\mathrm{e}}^{-nx}}.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $x$: real variable, $n$: nonnegative integer and $S_n(x)$: function

Referenced by:

§8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E15

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Annotations for §8.11(v), §8.11 and Ch.8

… ►

28: 13.6 Relations to Other Functions

… ►

§13.6(ii) Incomplete Gamma Functions

… ►When $a-b$ is an integer or $a$ is a positive integer the Kummer functions can be expressed as incomplete gamma functions (or generalized exponential integrals). … ►

13.6.5 $$M(a,a+1,-z)={\mathrm{e}}^{-z}M(1,a+1,z)=a{z}^{-a}\gamma (a,z),$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function and $z$: complex variable

A&S Ref:

13.6.10

Permalink:

http://dlmf.nist.gov/13.6.E5

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See also:

Annotations for §13.6(ii), §13.6 and Ch.13

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13.6.6 $$U(a,a,z)=z^{1-a}U(1,2-a,z)=z^{1-a}{\mathrm{e}}^z{E}_a\left(z\right)={\mathrm{e}}^z\mathrm{\Gamma }(1-a,z).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function and $z$: complex variable

A&S Ref:

13.6.28 (in different form)

Permalink:

http://dlmf.nist.gov/13.6.E6

Encodings:

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See also:

Annotations for §13.6(ii), §13.6 and Ch.13

…

29: Bibliography P

… ►

An asymptotic formula for the incomplete gamma function. Ž. Vyčisl. Mat. i Mat. Fiz. 5, pp. 118–121 (Russian).

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 5(1), pp. 162–166

External Links:

Document, MathReview (E. Lukacs), ZentralBlatt

Referenced by:

§8.11(iv).

Encodings:

BibTeX

… ►

Error bounds for the uniform asymptotic expansion of the incomplete gamma function. J. Comput. Appl. Math. 147 (1), pp. 215–231.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§8.12.

Encodings:

BibTeX

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A uniform asymptotic expansion for the incomplete gamma function. J. Comput. Appl. Math. 148 (2), pp. 323–339.

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External Links:

ISSN 0377-0427, Document, MathReview (Dumitru Acu), ZentralBlatt

Referenced by:

(8.11.6), §8.11(iii), (8.12.18), §8.12, §8.12, §8.12, Erratum (V1.0.16) for Equation (8.12.18).

Encodings:

BibTeX

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The asymptotic expansion of a generalised incomplete gamma function. J. Comput. Appl. Math. 151 (2), pp. 297–306.

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External Links:

ISSN 0377-0427, Document, MathReview (José Luis López), ZentralBlatt

Referenced by:

§8.16.

Encodings:

BibTeX

… ►

Tables of the Incomplete $\mathrm{\Gamma }$-function. Biometrika Office, Cambridge University Press, Cambridge.

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Referenced by:

3rd item.

Encodings:

BibTeX

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30: 8.28 Software

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§8.28(ii) Incomplete Gamma Functions for Real Argument and Parameter

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§8.28(iii) Incomplete Gamma Functions for Complex Argument and Parameter

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