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31: 15.2 Definitions and Analytical Properties

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15.2.1 $$F(a,b;c;z)=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s{\left(b\right)}_s}{{\left(c\right)}_{s}s!}{z}^s=1+\frac{ab}{c}z+\frac{a(a+1)b(b+1)}{c(c+1)2!}{z}^2+\mathrm{\cdots }=\frac{\mathrm{\Gamma }\left(c\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}\sum _{s=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a+s\right)\mathrm{\Gamma }\left(b+s\right)}{\mathrm{\Gamma }\left(c+s\right)s!}{z}^s,$$

ⓘ

Defines:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma 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), $!$: factorial (as in $n!$), $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

A&S Ref:

15.1.1

Referenced by:

§15.19(i), §15.4(iii), §18.30(v), §18.35(i), §18.35(ii), §19.19

Permalink:

http://dlmf.nist.gov/15.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.2(i), §15.2 and Ch.15

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15.2.2 $$𝐅(a,b;c;z)=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s{\left(b\right)}_s}{\mathrm{\Gamma }\left(c+s\right)s!}{z}^s,$$ ,

ⓘ

Defines:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.12(i), §16.2(v)

Permalink:

http://dlmf.nist.gov/15.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.2(i), §15.2 and Ch.15

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15.2.3_5 $$\underset{c\to -n}{lim}\frac{F(a,b;c;z)}{\mathrm{\Gamma }\left(c\right)}=𝐅(a,b;-n;z)=\frac{{\left(a\right)}_{n+1}{\left(b\right)}_{n+1}}{(n+1)!}{z}^{n+1}F(a+n+1,b+n+1;n+2;z),$$ $n=0,1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $n$: integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.2(ii), Erratum (V1.0.28) for Equations (15.2.3_5), (19.11.6_5)

Permalink:

http://dlmf.nist.gov/15.2.E3_5

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.28):

This equation was given a number.

See also:

Annotations for §15.2(ii), §15.2 and Ch.15

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15.2.5 $$F(\genfrac{}{}{0pt}{}{-m,b}{-m-\mathrm{\ell }};z)=\underset{c\to -m-\mathrm{\ell }}{lim}\left(\underset{a\to -m}{lim}F(\genfrac{}{}{0pt}{}{a,b}{c};z)\right),$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\mathrm{\ell }$: integer and $m$: integer

Referenced by:

§15.10(i), §15.2(ii), §15.2(ii), §15.4(i), §15.4(iii)

Permalink:

http://dlmf.nist.gov/15.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §15.2(ii), §15.2 and Ch.15

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15.2.6 $$F(\genfrac{}{}{0pt}{}{-m,b}{-m-\mathrm{\ell }};z)=\underset{a\to -m}{lim}F(\genfrac{}{}{0pt}{}{a,b}{a-\mathrm{\ell }};z),$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $\mathrm{\ell }$: integer and $m$: integer

Referenced by:

§15.2(ii), §15.2(ii), §15.4(i), §15.4(iii), Erratum (V1.0.16) for Subsection 15.2(ii)

Permalink:

http://dlmf.nist.gov/15.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §15.2(ii), §15.2 and Ch.15

…

32: 16.10 Expansions in Series of ${}_p{}^{}F_q^{}$ Functions

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16.10.1 $${}_{p+r}{}^{}F_{q+s}^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p,c_1,\mathrm{\dots },c_r}{b_1,\mathrm{\dots },b_q,d_1,\mathrm{\dots },d_s};z\zeta )=\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(𝐚\right)}_k{\left(\alpha \right)}_k{\left(\beta \right)}_k{(-z)}^k}{{\left(𝐛\right)}_k{\left(\gamma +k\right)}_{k}k!}{}_{p+2}{}^{}F_{q+1}^{}(\genfrac{}{}{0pt}{}{\alpha +k,\beta +k,a_1+k,\mathrm{\dots },a_p+k}{\gamma +2k+1,b_1+k,\mathrm{\dots },b_q+k};z){}_{r+2}{}^{}F_{s+2}^{}(\genfrac{}{}{0pt}{}{-k,\gamma +k,c_1,\mathrm{\dots },c_r}{\alpha ,\beta ,d_1,\mathrm{\dots },d_s};\zeta ).$$

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.10 and Ch.16

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16.10.2 $${}_{p+1}{}^{}F_p^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{p+1}}{b_1,\mathrm{\dots },b_p};z\zeta )={(1-z)}^{-a_1}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a_1\right)}_k}{k!}{}_{p+1}{}^{}F_p^{}(\genfrac{}{}{0pt}{}{-k,a_2,\mathrm{\dots },a_{p+1}}{b_1,\mathrm{\dots },b_p};\zeta ){\left(\frac{z}{z-1}\right)}^k.$$

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $p$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.10 and Ch.16

…

33: 8.1 Special Notation

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$x$	real variable.
$z$	complex variable.
$a,p$	real or complex parameters.
…

… ►The functions treated in this chapter are the incomplete gamma functions $\gamma (a,z)$, $\mathrm{\Gamma }(a,z)$, ${\gamma }^{\ast }(a,z)$, $P(a,z)$, and $Q(a,z)$; the incomplete beta functions ${\mathrm{B}}_x(a,b)$ and $I_x(a,b)$; the generalized exponential integral $E_p\left(z\right)$; the generalized sine and cosine integrals $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$. ►Alternative notations include: Prym’s functions $P_z(a)=\gamma (a,z)$, $Q_z(a)=\mathrm{\Gamma }(a,z)$, Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); $(a,z)!=\gamma (a+1,z)$, $[a,z]!=\mathrm{\Gamma }(a+1,z)$, Dingle (1973); $B(a,b,x)={\mathrm{B}}_x(a,b)$, $I(a,b,x)=I_x(a,b)$, Magnus et al. (1966); $\mathrm{Si}(a,x)\to \mathrm{Si}(1-a,x)$, $\mathrm{Ci}(a,x)\to \mathrm{Ci}(1-a,x)$, Luke (1975).

34: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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35.7.4 $$\underset{c\to \mathrm{\infty }}{lim}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};𝐈-c{𝐓}^{-1})={\left|𝐓\right|}^b\mathrm{\Psi }(b;b-a+\frac{1}{2}(m+1);𝐓).$$

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\mathrm{\Psi }(a;b;𝐓)$: confluent hypergeometric function of matrix argument (second kind), $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $b$: complex variable, $c$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

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35.7.7 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};𝐈)=\frac{{\mathrm{\Gamma }}_m\left(c\right){\mathrm{\Gamma }}_m\left(c-a-b\right)}{{\mathrm{\Gamma }}_m\left(c-a\right){\mathrm{\Gamma }}_m\left(c-b\right)},$$ $\mathrm{\Re }\left(c\right),\mathrm{\Re }\left(c-a-b\right)>\frac{1}{2}(m-1)$.

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $b$: complex variable, $c$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

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35.7.9 $$t_j(1-t_j)\frac{{\partial }^{2}F}{{\partial t_j}^2}-\frac{1}{2}\sum _{\begin{array}{c}k=1\\ k\ne j\end{array}}^m\frac{t_k(1-t_k)}{t_j-t_k}\frac{\partial F}{\partial t_k}+\left(c-\frac{1}{2}(m-1)-\left(a+b-\frac{1}{2}(m-3)\right)t_j+\frac{1}{2}\sum _{\begin{array}{c}k=1\\ k\ne j\end{array}}^m\frac{t_j(1-t_j)}{t_j-t_k}\right)\frac{\partial F}{\partial t_j}=abF,$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $a$: complex variable, $t_j$: eigenvalues of $𝐓$ , $b$: complex variable, $c$: complex variable, $j$: nonnegative integer, $k$: nonnegative integer and $m$: positive integer

Referenced by:

§35.7(iii)

Permalink:

http://dlmf.nist.gov/35.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(iii), §35.7 and Ch.35

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35.7.10 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha a,\alpha b}{\alpha c};𝐓)$$

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $c$: complex variable and $\alpha \to \mathrm{\infty }$: parameter

Permalink:

http://dlmf.nist.gov/35.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(iv), §35.7 and Ch.35

… ►

35.7.11 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};𝐈-{\alpha }^{-1}𝐓)$$

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $c$: complex variable and $\alpha \to \mathrm{\infty }$: parameter

Permalink:

http://dlmf.nist.gov/35.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(iv), §35.7 and Ch.35

…

35: 5.13 Integrals

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5.13.1 $$\frac{1}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(s+a\right)\mathrm{\Gamma }\left(b-s\right)z^{-s}ds=\frac{\mathrm{\Gamma }\left(a+b\right)z^a}{{(1+z)}^{a+b}},$$ $\mathrm{\Re }\left(a+b\right)>0$, , .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $s$: real or complex variable, $z$: complex variable, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.13, §5.13, §5.19(ii)

Permalink:

http://dlmf.nist.gov/5.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §5.13 and Ch.5

►

5.13.2 $$\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|\mathrm{\Gamma }\left(a+\mathrm{i}t\right)|}^2{\mathrm{e}}^{(2b-\pi )t}dt=\frac{\mathrm{\Gamma }\left(2a\right)}{{(2\sin b)}^{2a}},$$ $a>0$, .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.13

Permalink:

http://dlmf.nist.gov/5.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.13 and Ch.5

… ►

5.13.3 $$\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{\Gamma }\left(a+\mathrm{i}t\right)\mathrm{\Gamma }\left(b+\mathrm{i}t\right)\mathrm{\Gamma }\left(c-\mathrm{i}t\right)\mathrm{\Gamma }\left(d-\mathrm{i}t\right)dt=\frac{\mathrm{\Gamma }\left(a+c\right)\mathrm{\Gamma }\left(a+d\right)\mathrm{\Gamma }\left(b+c\right)\mathrm{\Gamma }\left(b+d\right)}{\mathrm{\Gamma }\left(a+b+c+d\right)},$$ $\mathrm{\Re }a,\mathrm{\Re }b,\mathrm{\Re }c,\mathrm{\Re }d>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.13

Permalink:

http://dlmf.nist.gov/5.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.13, §5.13 and Ch.5

… ►

5.13.4 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{dt}{\mathrm{\Gamma }\left(a+t\right)\mathrm{\Gamma }\left(b+t\right)\mathrm{\Gamma }\left(c-t\right)\mathrm{\Gamma }\left(d-t\right)}=\frac{\mathrm{\Gamma }\left(a+b+c+d-3\right)}{\mathrm{\Gamma }\left(a+c-1\right)\mathrm{\Gamma }\left(a+d-1\right)\mathrm{\Gamma }\left(b+c-1\right)\mathrm{\Gamma }\left(b+d-1\right)},$$ $\mathrm{\Re }\left(a+b+c+d\right)>3$.

ⓘ

Symbols:

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

Referenced by:

§5.13

Permalink:

http://dlmf.nist.gov/5.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §5.13, §5.13 and Ch.5

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5.13.5 $\mathrm{\Re }\left(a_k\right)>0$, $k=1,2,3,4$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $j$: nonnegative integer, $k$: nonnegative integer and $a$: real or complex variable

Referenced by:

§5.13

Permalink:

http://dlmf.nist.gov/5.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.13, §5.13 and Ch.5

…

36: 35.1 Special Notation

… ► ►►

$a,b$	complex variables.
…

…

37: 23.18 Modular Transformations

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23.18.3 $$\lambda \left(𝒜\tau \right)=\lambda \left(\tau \right),$$

ⓘ

Symbols:

$\lambda \left(\tau \right)$: elliptic modular function, $\tau$: complex variable and $𝒜$: bilinear transformation

Permalink:

http://dlmf.nist.gov/23.18.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.18, §23.18 and Ch.23

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23.18.4 $$J\left(𝒜\tau \right)=J\left(\tau \right).$$

ⓘ

Symbols:

$J\left(\tau \right)$: Klein’s complete invariant, $\tau$: complex variable and $𝒜$: bilinear transformation

Referenced by:

§23.18

Permalink:

http://dlmf.nist.gov/23.18.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §23.18, §23.18 and Ch.23

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23.18.5 $$\eta \left(𝒜\tau \right)=\epsilon (𝒜){\left(-\mathrm{i}(c\tau +d)\right)}^{1/2}\eta \left(\tau \right),$$

ⓘ

Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\mathrm{i}$: imaginary unit, $\tau$: complex variable, $𝒜$: bilinear transformation, $c$: integer, $d$: integer and $\epsilon (𝒜)$: function

Permalink:

http://dlmf.nist.gov/23.18.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.18, §23.18 and Ch.23

…

38: 5.11 Asymptotic Expansions

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5.11.7 $$\mathrm{\Gamma }\left(az+b\right)\sim \sqrt{2\pi }{\mathrm{e}}^{-az}{(az)}^{az+b-(1/2)},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $z$: complex variable, $b$: real or complex variable and $a_k$: coefficient

A&S Ref:

6.1.39

Referenced by:

§11.6(ii), §13.21(i), §2.10(iii), §2.10(iv), §26.3(v), §33.5(iv), §5.11(i)

Permalink:

http://dlmf.nist.gov/5.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.11(i), §5.11(i), §5.11 and Ch.5

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5.11.12 $$\frac{\mathrm{\Gamma }\left(z+a\right)}{\mathrm{\Gamma }\left(z+b\right)}\sim z^{a-b},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality, $z$: complex variable, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.11(i)

Permalink:

http://dlmf.nist.gov/5.11.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §5.11(iii), §5.11 and Ch.5

►

5.11.13 $$\frac{\mathrm{\Gamma }\left(z+a\right)}{\mathrm{\Gamma }\left(z+b\right)}\sim z^{a-b}\sum _{k=0}^{\mathrm{\infty }}\frac{G_k(a,b)}{z^k},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $z$: complex variable, $a$: real or complex variable, $b$: real or complex variable and $G_k(a,b)$: coefficients

A&S Ref:

6.1.47

Permalink:

http://dlmf.nist.gov/5.11.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §5.11(iii), §5.11 and Ch.5

►

5.11.14 $$\frac{\mathrm{\Gamma }\left(z+a\right)}{\mathrm{\Gamma }\left(z+b\right)}\sim {\left(z+\frac{a+b-1}{2}\right)}^{a-b}\sum _{k=0}^{\mathrm{\infty }}\frac{H_k(a,b)}{{\left(z+\frac{1}{2}(a+b-1)\right)}^{2k}}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $z$: complex variable, $a$: real or complex variable, $b$: real or complex variable and $H_k(a,b)$: coefficients

Referenced by:

§5.11(iii), Erratum (V1.0.21) for Equation (5.11.14)

Permalink:

http://dlmf.nist.gov/5.11.E14

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.21):

The previous constraint $\mathrm{\Re }\left(b-a\right)>0$ was removed, see Fields (1966, (3)).

See also:

Annotations for §5.11(iii), §5.11 and Ch.5

… ►

5.11.19 $$\frac{\mathrm{\Gamma }\left(z+a\right)\mathrm{\Gamma }\left(z+b\right)}{\mathrm{\Gamma }\left(z+c\right)}\sim \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{{\left(c-a\right)}_k{\left(c-b\right)}_k}{k!}\mathrm{\Gamma }\left(a+b-c+z-k\right).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $k$: nonnegative integer, $z$: complex variable, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.11(i), §5.11(iii)

Permalink:

http://dlmf.nist.gov/5.11.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §5.11(iii), §5.11 and Ch.5

…

39: 16.11 Asymptotic Expansions

… ►

16.11.2 $$H_{p,q}(z)=\sum _{m=1}^p\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k}{k!}\mathrm{\Gamma }\left(a_m+k\right)\left(\prod _{\begin{array}{c}\mathrm{\ell }=1\\ \mathrm{\ell }\ne m\end{array}}^p\mathrm{\Gamma }\left(a_{\mathrm{\ell }}-a_m-k\right)/\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-a_m-k\right)\right)z^{-a_m-k}.$$

ⓘ

Defines:

$H_{p,q}(z)$: formal infinite series (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.11(ii), §16.11(ii)

Permalink:

http://dlmf.nist.gov/16.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(i), §16.11 and Ch.16

… ►

16.11.6 $$\left(\prod _{\mathrm{\ell }=1}^{q+1}\mathrm{\Gamma }\left(a_{\mathrm{\ell }}\right)/\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(b_{\mathrm{\ell }}\right)\right){}_{q+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{q+1}}{b_1,\mathrm{\dots },b_q};z)=H_{q+1,q}(-z),$$ $|\mathrm{ph}\left(-z\right)|\le \pi$;

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $H_{p,q}(z)$: formal infinite series

Permalink:

http://dlmf.nist.gov/16.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

… ►

16.11.8 $$\left(\prod _{\mathrm{\ell }=1}^{q-1}\mathrm{\Gamma }\left(a_{\mathrm{\ell }}\right)/\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(b_{\mathrm{\ell }}\right)\right){}_{q-1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{q-1}}{b_1,\mathrm{\dots },b_q};-z)\sim H_{q-1,q}(z)+E_{q-1,q}(z{e}^{-\pi \mathrm{i}})+E_{q-1,q}(z{e}^{\pi \mathrm{i}}),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $E_{p,q}(z)$: formal infinite series, $H_{p,q}(z)$: formal infinite series and $e_{k,m}$

Permalink:

http://dlmf.nist.gov/16.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

… ►

16.11.9 $$\left(\prod _{\mathrm{\ell }=1}^p\mathrm{\Gamma }\left(a_{\mathrm{\ell }}\right)/\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(b_{\mathrm{\ell }}\right)\right){}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};-z)\sim E_{p,q}(z{e}^{-\pi \mathrm{i}})+E_{p,q}(z{e}^{\pi \mathrm{i}}),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $E_{p,q}(z)$: formal infinite series and $e_{k,m}$

Referenced by:

§16.11(ii)

Permalink:

http://dlmf.nist.gov/16.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

… ►

16.11.11 $${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1+r,\mathrm{\dots },a_p+r}{b_1+r,\mathrm{\dots },b_q+r};z)=\sum _{n=0}^{m-1}\frac{{\left(a_1+r\right)}_n\mathrm{\cdots }{\left(a_p+r\right)}_n}{{\left(b_1+r\right)}_n\mathrm{\cdots }{\left(b_q+r\right)}_n}\frac{z^n}{n!}+O\left(\frac{1}{r^{(q-p)m}}\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $r$: large parameter

Permalink:

http://dlmf.nist.gov/16.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(iii), §16.11 and Ch.16

…

40: 16.8 Differential Equations

… ►

16.8.3 $$\left(\mathit{\vartheta }(\mathit{\vartheta }+b_1-1)\mathrm{\cdots }(\mathit{\vartheta }+b_q-1)-z(\mathit{\vartheta }+a_1)\mathrm{\cdots }(\mathit{\vartheta }+a_p)\right)w=0.$$

ⓘ

Symbols:

$p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $\mathit{\vartheta }$: differential operator and $w$: solution

Referenced by:

§16.8(ii), §16.8(ii), §16.8(ii)

Permalink:

http://dlmf.nist.gov/16.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(ii), §16.8 and Ch.16

… ►

16.8.7 $${\tilde{w}}_j(z)={(-z)}^{-a_j}{}_{q+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_j,1-b_1+a_j,\mathrm{\dots },1-b_q+a_j}{1-a_1+a_j,\mathrm{\dots }\ast \mathrm{\dots },1-a_{q+1}+a_j};\frac{1}{z}),$$ $j=1,\mathrm{\dots },q+1$,

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $w$: solution

Permalink:

http://dlmf.nist.gov/16.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(ii), §16.8 and Ch.16

… ►

16.8.8 $${}_{q+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{q+1}}{b_1,\mathrm{\dots },b_q};z)=\sum _{j=1}^{q+1}\left(\prod _{\begin{array}{c}k=1\\ k\ne j\end{array}}^{q+1}\frac{\mathrm{\Gamma }\left(a_k-a_j\right)}{\mathrm{\Gamma }\left(a_k\right)}/\prod _{k=1}^q\frac{\mathrm{\Gamma }\left(b_k-a_j\right)}{\mathrm{\Gamma }\left(b_k\right)}\right){\tilde{w}}_j(z),$$ $|\mathrm{ph}\left(-z\right)|\le \pi$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $w$: solution

Referenced by:

§16.11(ii)

Permalink:

http://dlmf.nist.gov/16.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(ii), §16.8 and Ch.16

… ►

16.8.10 $$\underset{|\alpha |\to \mathrm{\infty }}{lim}{}_{p+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p,\alpha }{b_1,\mathrm{\dots },b_q};\frac{z}{\alpha })={}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z).$$

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.8.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(iii), §16.8 and Ch.16

… ►

16.8.11 $$\underset{|\beta |\to \mathrm{\infty }}{lim}{}_p{}^{}F_{q+1}^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q,\beta };\beta z)={}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z),$$

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(iii), §16.8 and Ch.16

…