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1: 16.15 Integral Representations and Integrals

… ►

16.15.1 $$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)=\frac{\mathrm{\Gamma }\left(\gamma \right)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(\gamma -\alpha \right)}{\int }_0^1\frac{u^{\alpha -1}{(1-u)}^{\gamma -\alpha -1}}{{(1-ux)}^{\beta }{(1-uy)}^{{\beta }^{\prime }}}du,$$ $\mathrm{\Re }\alpha >0$, $\mathrm{\Re }\left(\gamma -\alpha \right)>0$,

ⓘ

Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/16.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.15 and Ch.16

►

16.15.2 $$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)=\frac{\mathrm{\Gamma }\left(\gamma \right)\mathrm{\Gamma }\left({\gamma }^{\prime }\right)}{\mathrm{\Gamma }\left(\beta \right)\mathrm{\Gamma }\left({\beta }^{\prime }\right)\mathrm{\Gamma }\left(\gamma -\beta \right)\mathrm{\Gamma }\left({\gamma }^{\prime }-{\beta }^{\prime }\right)}{\int }_0^1{\int }_0^1\frac{u^{\beta -1}{v}^{{\beta }^{\prime }-1}{(1-u)}^{\gamma -\beta -1}{(1-v)}^{{\gamma }^{\prime }-{\beta }^{\prime }-1}}{{(1-ux-vy)}^{\alpha }}dudv,$$ $\mathrm{\Re }\gamma >\mathrm{\Re }\beta >0$, $\mathrm{\Re }{\gamma }^{\prime }>\mathrm{\Re }{\beta }^{\prime }>0$,

ⓘ

Symbols:

$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$: second Appell function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/16.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.15 and Ch.16

►

16.15.3 $$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)=\frac{\mathrm{\Gamma }\left(\gamma \right)}{\mathrm{\Gamma }\left(\beta \right)\mathrm{\Gamma }\left({\beta }^{\prime }\right)\mathrm{\Gamma }\left(\gamma -\beta -{\beta }^{\prime }\right)}{\iint }_{\mathrm{\Delta }}\frac{u^{\beta -1}{v}^{{\beta }^{\prime }-1}{(1-u-v)}^{\gamma -\beta -{\beta }^{\prime }-1}}{{(1-ux)}^{\alpha }{(1-vy)}^{{\alpha }^{\prime }}}dudv,$$ $\mathrm{\Re }\left(\gamma -\beta -{\beta }^{\prime }\right)>0$, $\mathrm{\Re }\beta >0$, $\mathrm{\Re }{\beta }^{\prime }>0$,

ⓘ

Symbols:

$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$: third Appell function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Notes:

Erdélyi et al. (1953a, 5.8.1(3)) contains a typo.

Referenced by:

Erratum (V1.0.14) for Equation (16.15.3), Erratum (V1.0.14) for Equation (16.15.3)

Permalink:

http://dlmf.nist.gov/16.15.E3

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.14):

A stray “$f$” was removed; it appeared after the comma at the end of this equation.

See also:

Annotations for §16.15 and Ch.16

… ►

16.15.4 $$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x(1-y),y(1-x))=\frac{\mathrm{\Gamma }\left(\gamma \right)\mathrm{\Gamma }\left({\gamma }^{\prime }\right)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(\beta \right)\mathrm{\Gamma }\left(\gamma -\alpha \right)\mathrm{\Gamma }\left({\gamma }^{\prime }-\beta \right)}{\int }_0^1{\int }_0^1\frac{u^{\alpha -1}{v}^{\beta -1}{(1-u)}^{\gamma -\alpha -1}{(1-v)}^{{\gamma }^{\prime }-\beta -1}}{{(1-ux)}^{\gamma +{\gamma }^{\prime }-\alpha -1}{(1-vy)}^{\gamma +{\gamma }^{\prime }-\beta -1}{(1-ux-vy)}^{\alpha +\beta -\gamma -{\gamma }^{\prime }+1}}dudv,$$ $\mathrm{\Re }\gamma >\mathrm{\Re }\alpha >0$, $\mathrm{\Re }{\gamma }^{\prime }>\mathrm{\Re }\beta >0$.

ⓘ

Symbols:

$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Source:

Burchnall and Chaundy (1940, (68))

Referenced by:

§16.15

Permalink:

http://dlmf.nist.gov/16.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §16.15 and Ch.16

…

2: 30.1 Special Notation

… ► ►►►

$x$	real variable. Except in §§30.7(iv), 30.11(ii), 30.13, and 30.14, .
${\gamma }^2$	real parameter (positive, zero, or negative).
…

►The main functions treated in this chapter are the eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$ and the spheroidal wave functions ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$, ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$, ${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)$, and $S_n^{m(j)}(z,\gamma )$, $j=1,2,3,4$. …Meixner and Schäfke (1954) use $\mathrm{ps}$, $\mathrm{qs}$, $\mathrm{Ps}$, $\mathrm{Qs}$ for $\mathrm{𝖯𝗌}$, $\mathrm{𝖰𝗌}$, $\mathrm{𝑃𝑠}$, $\mathrm{𝑄𝑠}$, respectively. … ►Flammer (1957) and Abramowitz and Stegun (1964) use ${\lambda }_{mn}(\gamma )$ for ${\lambda }_n^m\left({\gamma }^2\right)+{\gamma }^2$, $R_{mn}^{(j)}(\gamma ,z)$ for $S_n^{m(j)}(z,\gamma )$, and …where $d_{mn}(\gamma )$ is a normalization constant determined by …

3: 30.5 Functions of the Second Kind

… ►Other solutions of (30.2.1) with $\mu =m$, $\lambda ={\lambda }_n^m\left({\gamma }^2\right)$, and $z=x$ are ►

30.5.1 $${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2),$$ $n=m,m+1,m+2,\mathrm{\dots }$.

ⓘ

Symbols:

${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §30.5 and Ch.30

… ►

30.5.2 $${\mathrm{𝖰𝗌}}_n^m(-x,{\gamma }^2)={(-1)}^{n-m+1}{\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2),$$

ⓘ

Symbols:

${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §30.5 and Ch.30

… ►

30.5.4 $$𝒲\left\{{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2),{\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)\right\}=\frac{(n+m)!}{(1-x^2)(n-m)!}{A}_n^m({\gamma }^2)A_n^{-m}({\gamma }^2)(\ne 0),$$

ⓘ

Symbols:

$𝒲$: Wronskian, $!$: factorial (as in $n!$), ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, $A_n^m({\gamma }^2)$ and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.5 and Ch.30

►with $A_n^{\pm m}({\gamma }^2)$ as in (30.11.4). …

4: 16.16 Transformations of Variables

… ►

16.16.5 $$F_3(\alpha ,\gamma -\alpha ;\beta ,\gamma -\beta ;\gamma ;x,y)={(1-y)}^{\alpha +\beta -\gamma }{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha ,\beta }{\gamma };x+y-xy),$$

ⓘ

Symbols:

$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$: third Appell function and ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function

Permalink:

http://dlmf.nist.gov/16.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §16.16(i), §16.16 and Ch.16

►

16.16.5_5 $$F_4(\alpha ,\beta ;\gamma ,\beta ;x(1-y),y(1-x))={(1-x)}^{-\alpha }{(1-y)}^{-\alpha }{F}_1(\alpha ;\gamma -\beta ,\alpha -\gamma +1;\gamma ;\frac{x}{x-1},\frac{xy}{(1-x)(1-y)}),$$

ⓘ

Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function and $F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function

Referenced by:

§16.16(i), Erratum (V1.1.12) for Additions

Permalink:

http://dlmf.nist.gov/16.16.E5_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.12):

This equation was added.

See also:

Annotations for §16.16(i), §16.16 and Ch.16

… ►

16.16.7 $$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x(1-y),y(1-x))=\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_k{\left(\beta \right)}_k{\left(\alpha +\beta -\gamma -{\gamma }^{\prime }+1\right)}_k}{{\left(\gamma \right)}_k{\left({\gamma }^{\prime }\right)}_{k}k!}{x}^k{y}^k{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha +k,\beta +k}{\gamma +k};x){}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha +k,\beta +k}{{\gamma }^{\prime }+k};y);$$

ⓘ

Symbols:

$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.16.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §16.16(i), §16.16 and Ch.16

… ►

16.16.9 $$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)={(1-x)}^{-\alpha }{F}_2(\alpha ;\gamma -\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };\frac{x}{x-1},\frac{y}{1-x}),$$

ⓘ

Symbols:

$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$: second Appell function

Permalink:

http://dlmf.nist.gov/16.16.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §16.16(ii), §16.16 and Ch.16

►

16.16.10 $$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)=\frac{\mathrm{\Gamma }\left({\gamma }^{\prime }\right)\mathrm{\Gamma }\left(\beta -\alpha \right)}{\mathrm{\Gamma }\left({\gamma }^{\prime }-\alpha \right)\mathrm{\Gamma }\left(\beta \right)}{(-y)}^{-\alpha }{F}_4(\alpha ,\alpha -{\gamma }^{\prime }+1;\gamma ,\alpha -\beta +1;\frac{x}{y},\frac{1}{y})+\frac{\mathrm{\Gamma }\left({\gamma }^{\prime }\right)\mathrm{\Gamma }\left(\alpha -\beta \right)}{\mathrm{\Gamma }\left({\gamma }^{\prime }-\beta \right)\mathrm{\Gamma }\left(\alpha \right)}{(-y)}^{-\beta }{F}_4(\beta ,\beta -{\gamma }^{\prime }+1;\gamma ,\beta -\alpha +1;\frac{x}{y},\frac{1}{y}).$$

ⓘ

Symbols:

$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function and $\mathrm{\Gamma }\left(z\right)$: gamma function

Permalink:

http://dlmf.nist.gov/16.16.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §16.16(ii), §16.16 and Ch.16

…

5: 5.22 Tables

… ►Abramowitz and Stegun (1964, Chapter 6) tabulates $\mathrm{\Gamma }\left(x\right)$, $\ln \mathrm{\Gamma }\left(x\right)$, $\psi \left(x\right)$, and ${\psi }^{\prime }\left(x\right)$ for $x=1(.005)2$ to 10D; ${\psi }^{\prime \prime }\left(x\right)$ and ${\psi }^{(3)}\left(x\right)$ for $x=1(.01)2$ to 10D; $\mathrm{\Gamma }\left(n\right)$, $1/\mathrm{\Gamma }\left(n\right)$, $\mathrm{\Gamma }\left(n+\frac{1}{2}\right)$, $\psi \left(n\right)$, ${\log }_{10}\mathrm{\Gamma }\left(n\right)$, ${\log }_{10}\mathrm{\Gamma }\left(n+\frac{1}{3}\right)$, ${\log }_{10}\mathrm{\Gamma }\left(n+\frac{1}{2}\right)$, and ${\log }_{10}\mathrm{\Gamma }\left(n+\frac{2}{3}\right)$ for $n=1(1)101$ to 8–11S; $\mathrm{\Gamma }\left(n+1\right)$ for $n=100(100)1000$ to 20S. Zhang and Jin (1996, pp. 67–69 and 72) tabulates $\mathrm{\Gamma }\left(x\right)$, $1/\mathrm{\Gamma }\left(x\right)$, $\mathrm{\Gamma }\left(-x\right)$, $\ln \mathrm{\Gamma }\left(x\right)$, $\psi \left(x\right)$, $\psi \left(-x\right)$, ${\psi }^{\prime }\left(x\right)$, and ${\psi }^{\prime }\left(-x\right)$ for $x=0(.1)5$ to 8D or 8S; $\mathrm{\Gamma }\left(n+1\right)$ for $n=0(1)100(10)250(50)500(100)3000$ to 51S. … ►Abramov (1960) tabulates $\ln \mathrm{\Gamma }\left(x+\mathrm{i}y\right)$ for $x=1$ ($.01$) $2$, $y=0$ ($.01$) $4$ to 6D. Abramowitz and Stegun (1964, Chapter 6) tabulates $\ln \mathrm{\Gamma }\left(x+\mathrm{i}y\right)$ for $x=1$ ($.1$) $2$, $y=0$ ($.1$) $10$ to 12D. …Zhang and Jin (1996, pp. 70, 71, and 73) tabulates the real and imaginary parts of $\mathrm{\Gamma }\left(x+\mathrm{i}y\right)$, $\ln \mathrm{\Gamma }\left(x+\mathrm{i}y\right)$, and $\psi \left(x+\mathrm{i}y\right)$ for $x=0.5,1,5,10$, $y=0(.5)10$ to 8S.

6: 30.6 Functions of Complex Argument

… ►The solutions ►

${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2),$

►

${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2),$

… ►

30.6.3 $$𝒲\left\{{\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2),{\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)\right\}=\frac{{(-1)}^m(n+m)!}{(1-z^2)(n-m)!}{A}_n^m({\gamma }^2)A_n^{-m}({\gamma }^2),$$

ⓘ

Symbols:

$𝒲$: Wronskian, $!$: factorial (as in $n!$), ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, ${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, $z$: complex variable, $m$: nonnegative integer, $n\ge m$: integer degree, $A_n^m({\gamma }^2)$ and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.6, §30.6 and Ch.30

►with $A_n^{\pm m}({\gamma }^2)$ as in (30.11.4). …

7: 8.8 Recurrence Relations and Derivatives

… ►

8.8.1 $$\gamma (a+1,z)=a\gamma (a,z)-z^a{\mathrm{e}}^{-z},$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.22

Permalink:

http://dlmf.nist.gov/8.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

… ►If $w(a,z)=\gamma (a,z)$ or $\mathrm{\Gamma }(a,z)$, then … ►

8.8.4 $$z{\gamma }^{\ast }(a+1,z)={\gamma }^{\ast }(a,z)-\frac{{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a+1\right)}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

… ►

8.8.8 $$\gamma (a,z)=\frac{\mathrm{\Gamma }\left(a\right)}{\mathrm{\Gamma }\left(a-n\right)}\gamma (a-n,z)-z^{a-1}{\mathrm{e}}^{-z}\sum _{k=0}^{n-1}\frac{\mathrm{\Gamma }\left(a\right)}{\mathrm{\Gamma }\left(a-k\right)}{z}^{-k},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/8.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

… ►

8.8.10 $$\mathrm{\Gamma }(a,z)=\frac{\mathrm{\Gamma }\left(a\right)}{\mathrm{\Gamma }\left(a-n\right)}\mathrm{\Gamma }(a-n,z)+z^{a-1}{\mathrm{e}}^{-z}\sum _{k=0}^{n-1}\frac{\mathrm{\Gamma }\left(a\right)}{\mathrm{\Gamma }\left(a-k\right)}{z}^{-k},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/8.8.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

…

8: 30.4 Functions of the First Kind

… ►The eigenfunctions of (30.2.1) that correspond to the eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$ are denoted by ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$, $n=m,m+1,m+2,\mathrm{\dots }$. …the sign of ${\mathrm{𝖯𝗌}}_n^m(0,{\gamma }^2)$ being ${(-1)}^{(n+m)/2}$ when $n-m$ is even, and the sign of ${d{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)/dx|}_{x=0}$ being ${(-1)}^{(n+m-1)/2}$ when $n-m$ is odd. ►When ${\gamma }^2>0$ ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ is the prolate angular spheroidal wave function, and when ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ is the oblate angular spheroidal wave function. … ►

30.4.3 $${\mathrm{𝖯𝗌}}_n^m(-x,{\gamma }^2)={(-1)}^{n-m}{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2).$$

ⓘ

Symbols:

${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(ii), §30.4 and Ch.30

► ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ has exactly $n-m$ zeros in the interval . …

9: 8.2 Definitions and Basic Properties

… ►The general values of the incomplete gamma functions $\gamma (a,z)$ and $\mathrm{\Gamma }(a,z)$ are defined by … ►

8.2.3 $$\gamma (a,z)+\mathrm{\Gamma }(a,z)=\mathrm{\Gamma }\left(a\right),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\gamma (a,z)$: 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:

pMML, png, TeX

See also:

Annotations for §8.2(i), §8.2 and Ch.8

… ►In this subsection the functions $\gamma$ and $\mathrm{\Gamma }$ have their general values. ►The function ${\gamma }^{\ast }(a,z)$ is entire in $z$ and $a$. … ►If $w=\gamma (a,z)$ or $\mathrm{\Gamma }(a,z)$, then …

10: 5.8 Infinite Products

… ►

5.8.1 $$\mathrm{\Gamma }\left(z\right)=\underset{k\to \mathrm{\infty }}{lim}\frac{k!k^z}{z(z+1)\mathrm{\cdots }(z+k)},$$ $z\ne 0,-1,-2,\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.1.2

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §5.8 and Ch.5

►

5.8.2 $$\frac{1}{\mathrm{\Gamma }\left(z\right)}=z{\mathrm{e}}^{\gamma z}\prod _{k=1}^{\mathrm{\infty }}\left(1+\frac{z}{k}\right){\mathrm{e}}^{-z/k},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.1.3

Permalink:

http://dlmf.nist.gov/5.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.8 and Ch.5

►

5.8.3 $${\left|\frac{\mathrm{\Gamma }\left(x\right)}{\mathrm{\Gamma }\left(x+\mathrm{i}y\right)}\right|}^2=\prod _{k=0}^{\mathrm{\infty }}\left(1+\frac{y^2}{{(x+k)}^2}\right),$$ $x\ne 0,-1,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $x$: real variable and $y$: real variable

A&S Ref:

6.1.25 (where the formula for the reciprocal is given.)

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.8 and Ch.5

… ►

5.8.5 $$\prod _{k=0}^{\mathrm{\infty }}\frac{(a_1+k)(a_2+k)\mathrm{\cdots }(a_m+k)}{(b_1+k)(b_2+k)\mathrm{\cdots }(b_m+k)}=\frac{\mathrm{\Gamma }\left(b_1\right)\mathrm{\Gamma }\left(b_2\right)\mathrm{\cdots }\mathrm{\Gamma }\left(b_m\right)}{\mathrm{\Gamma }\left(a_1\right)\mathrm{\Gamma }\left(a_2\right)\mathrm{\cdots }\mathrm{\Gamma }\left(a_m\right)},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $m$: nonnegative integer, $k$: nonnegative integer, $a_k$: coefficient and $b_k$: coefficient

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.8 and Ch.5

…