Olver hypergeometric function

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1: 15.1 Special Notation

… ►and also ►

15.1.2

\frac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}=\mathbf{F}\left(a,b;c;z% \right)=\mathbf{F}\left({a,b\atop c};z\right)={{}_{2}{\mathbf{F}}_{1}}\left(a,% b;c;z\right),

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ or ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{% \mathbf{b}}};\NVar{z}\right)$ : scaled (or Olver’s) generalized hypergeometric function, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.1.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §15.1 and Ch.15

…

2: 15.6 Integral Representations

… ►

15.6.1

\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(b\right)\Gamma\left(c-b% \right)}\int_{0}^{1}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\,\mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

\Re c>\Re b>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Keywords:: Mellin transform
Referenced by:: §15.19(iii), (15.4.34), §15.6, §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9), Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)
Permalink:: http://dlmf.nist.gov/15.6.E1
Encodings:: pMML, png, TeX
Addition (effective with 1.0.23):: The Olverhypergeometricfunction $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
See also:: Annotations for §15.6 and Ch.15

►

15.6.2

\mathbf{F}\left(a,b;c;z\right)=\frac{\Gamma\left(1+b-c\right)}{2\pi\mathrm{i}% \Gamma\left(b\right)}\int_{0}^{(1+)}\frac{t^{b-1}(t-1)^{c-b-1}}{(1-zt)^{a}}\,% \mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

c-b\neq 1,2,3,\dots

\Re b>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Keywords:: Mellin transform
Referenced by:: §15.6
Permalink:: http://dlmf.nist.gov/15.6.E2
Encodings:: pMML, png, TeX
Addition (effective with 1.0.23):: The Olverhypergeometricfunction $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
See also:: Annotations for §15.6 and Ch.15

… ►

15.6.6

\mathbf{F}\left(a,b;c;z\right)=\frac{1}{2\pi\mathrm{i}\Gamma\left(a\right)% \Gamma\left(b\right)}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma% \left(a+t\right)\Gamma\left(b+t\right)\Gamma\left(-t\right)}{\Gamma\left(c+t% \right)}(-z)^{t}\,\mathrm{d}t,

|\operatorname{ph}\left(-z\right)|<\pi

;

a,b\neq 0,-1,-2,\dots

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : phase, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
A&S Ref:: 15.3.2 (modified)
Referenced by:: §15.6, §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)
Permalink:: http://dlmf.nist.gov/15.6.E6
Encodings:: pMML, png, TeX
Addition (effective with 1.0.23):: The Olverhypergeometricfunction $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
See also:: Annotations for §15.6 and Ch.15

►

15.6.7

\mathbf{F}\left(a,b;c;z\right)=\frac{1}{2\pi\mathrm{i}\Gamma\left(a\right)% \Gamma\left(b\right)\Gamma\left(c-a\right)\Gamma\left(c-b\right)}\int_{-% \mathrm{i}\infty}^{\mathrm{i}\infty}\Gamma\left(a+t\right)\Gamma\left(b+t% \right)\Gamma\left(c-a-b-t\right)\Gamma\left(-t\right)(1-z)^{t}\,\mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

a,b,c-a,c-b\neq 0,-1,-2,\dots

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : phase, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)
Permalink:: http://dlmf.nist.gov/15.6.E7
Encodings:: pMML, png, TeX
Addition (effective with 1.0.23):: The Olverhypergeometricfunction $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
See also:: Annotations for §15.6 and Ch.15

►

15.6.8

\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(c-d\right)}\int_{0}^{1}% \mathbf{F}\left(a,b;d;zt\right)t^{d-1}(1-t)^{c-d-1}\,\mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

\Re c>\Re d>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Keywords:: Mellin transform
Proof sketch:: Follows from (15.6.9) with $\lambda=a+b$ .
Referenced by:: (15.6.9), §15.6, §15.6, §15.6, §18.17(iv), Erratum (V1.0.14) for Equation (15.6.8), Erratum (V1.0.14) for Equation (15.6.8)
Permalink:: http://dlmf.nist.gov/15.6.E8
Encodings:: pMML, png, TeX
Addition (effective with 1.0.23):: The Olverhypergeometricfunction $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
See also:: Annotations for §15.6 and Ch.15

…

3: 13.1 Special Notation

… ►The main functions treated in this chapter are the Kummer functions

M\left(a,b,z\right)

and

U\left(a,b,z\right)

, Olver’s function

{\mathbf{M}}\left(a,b,z\right)

, and the Whittaker functions

M_{\kappa,\mu}\left(z\right)

and

W_{\kappa,\mu}\left(z\right)

. …

4: 14.3 Definitions and Hypergeometric Representations

… ►

14.3.3

\mathbf{F}\left(a,b;c;x\right)=\frac{1}{\Gamma\left(c\right)}F\left(a,b;c;x\right)

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function and $x$ : real variable $1<x<1$
Permalink:: http://dlmf.nist.gov/14.3.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §14.3(i), §14.3(i), §14.3 and Ch.14

►is Olver’s hypergeometric function (§15.1). … ►

14.3.6

P^{\mu}_{\nu}\left(x\right)=\left(\frac{x+1}{x-1}\right)^{\mu/2}\mathbf{F}% \left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right).

ⓘ

Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $\mu$ : general order, $\nu$ : general degree and $x$ : real variable $x>1$
Referenced by:: §14.21(i), §14.21(iii), §14.3(ii), §14.3(iv), §15.9(iv)
Permalink:: http://dlmf.nist.gov/14.3.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §14.3(ii), §14.3(ii), §14.3 and Ch.14

… ►

14.3.9

P^{-\mu}_{\nu}\left(x\right)=\left(\frac{x-1}{x+1}\right)^{\mu/2}\mathbf{F}% \left(\nu+1,-\nu;\mu+1;\tfrac{1}{2}-\tfrac{1}{2}x\right),

ⓘ

Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $\mu$ : general order, $\nu$ : general degree and $x$ : real variable $x>1$
Permalink:: http://dlmf.nist.gov/14.3.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §14.3(ii), §14.3(ii), §14.3 and Ch.14

… ►

14.3.15

P^{-\mu}_{\nu}\left(x\right)=2^{-\mu}\left(x^{2}-1\right)^{\mu/2}\mathbf{F}% \left(\mu-\nu,\nu+\mu+1;\mu+1;\tfrac{1}{2}-\tfrac{1}{2}x\right),

ⓘ

Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $\mu$ : general order, $\nu$ : general degree and $x$ : real variable $x>1$
Referenced by:: §14.21(iii)
Permalink:: http://dlmf.nist.gov/14.3.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §14.3(iii), §14.3 and Ch.14

…

5: 15.7 Continued Fractions

… ►

15.7.1

\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=t_{0% }-\cfrac{u_{1}z}{t_{1}-\cfrac{u_{2}z}{t_{2}-\cfrac{u_{3}z}{t_{3}-\cdots}}},

ⓘ

Symbols:: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $t_{n}$ : coefficient and $u_{n}$ : coefficient
Permalink:: http://dlmf.nist.gov/15.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.7 and Ch.15

… ►

15.7.3

\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=v_{0% }-\cfrac{w_{1}}{v_{1}-\cfrac{w_{2}}{v_{2}-\cfrac{w_{3}}{v_{3}-\cdots}}},

ⓘ

Symbols:: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $v_{n}$ : coefficient and $w_{n}$ : coefficient
Permalink:: http://dlmf.nist.gov/15.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §15.7 and Ch.15

… ►

15.7.5

\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a+1,b+1;c+1;z\right)}={x% _{0}+\cfrac{y_{1}}{x_{1}+\cfrac{y_{2}}{x_{2}+\cfrac{y_{3}}{x_{3}+\cdots}}}},

ⓘ

Symbols:: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $x$ : real variable, $y$ : real variable, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.7.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §15.7 and Ch.15

…

6: 15.15 Sums

… ►

15.15.1

\mathbf{F}\left({a,b\atop c};\frac{1}{z}\right)=\left(1-\frac{z_{0}}{z}\right)% ^{-a}\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}}{s!}\*\mathbf{F}\left({-s,b% \atop c};\frac{1}{z_{0}}\right)\left(1-\frac{z}{z_{0}}\right)^{-s}.

ⓘ

Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s 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:: §15.19(i)
Permalink:: http://dlmf.nist.gov/15.15.E1
Encodings:: pMML, png, TeX
Correction (effective with 1.1.2):: The Pochhammer symbol now links to its definition.
See also:: Annotations for §15.15 and Ch.15

…

7: 15.14 Integrals

… ►

15.14.1

\int_{0}^{\infty}x^{s-1}\mathbf{F}\left({a,b\atop c};-x\right)\,\mathrm{d}x=% \frac{\Gamma\left(s\right)\Gamma\left(a-s\right)\Gamma\left(b-s\right)}{\Gamma% \left(a\right)\Gamma\left(b\right)\Gamma\left(c-s\right)},

\min(\Re a,\Re b)>\Re s>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $x$ : real variable, $s$ : nonnegative integer, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Keywords:: Mellin transform
Referenced by:: §15.14
Permalink:: http://dlmf.nist.gov/15.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.14 and Ch.15

…

8: 15.2 Definitions and Analytical Properties

… ►

15.2.2

\mathbf{F}\left(a,b;c;z\right)=\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}{% \left(b\right)_{s}}}{\Gamma\left(c+s\right)s!}z^{s},

|z|<1

ⓘ

Defines:: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function
Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ or ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{% \mathbf{b}}};\NVar{z}\right)$ : 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

… ►

15.2.3

\mathbf{F}\left({a,b\atop c};x+\mathrm{i}0\right)-\mathbf{F}\left({a,b\atop c}% ;x-\mathrm{i}0\right)=\frac{2\pi\mathrm{i}}{\Gamma\left(a\right)\Gamma\left(b% \right)}(x-1)^{c-a-b}\mathbf{F}\left({c-a,c-b\atop c-a-b+1};1-x\right),

x>1

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $x$ : real variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.2(i)
Permalink:: http://dlmf.nist.gov/15.2.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §15.2(i), §15.2 and Ch.15

… ►

15.2.3_5

\lim_{c\to-n}\frac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}=\mathbf{F}% \left(a,b;-n;z\right)=\frac{{\left(a\right)_{n+1}}{\left(b\right)_{n+1}}}{(n+1% )!}z^{n+1}F\left(a+n+1,b+n+1;n+2;z\right),

n=0,1,2,\dots

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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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9: 13.10 Integrals

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13.10.1

\int{\mathbf{M}}\left(a,b,z\right)\,\mathrm{d}z=\frac{1}{a-1}{\mathbf{M}}\left% (a-1,b-1,z\right),

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Symbols:: ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §13.10(i), §13.10 and Ch.13

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13.10.3

\int_{0}^{\infty}e^{-zt}t^{b-1}{\mathbf{M}}\left(a,c,kt\right)\,\mathrm{d}t=% \Gamma\left(b\right)z^{-b}{{}_{2}{\mathbf{F}}_{1}}\left(a,b;c;\ifrac{k}{z}% \right),

\Re b>0

\Re z>\max\left(\Re k,0\right)

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ or ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{% \mathbf{b}}};\NVar{z}\right)$ : scaled (or Olver’s) generalized hypergeometric function, $\int$ : integral, $\Re$ : real part and $z$ : complex variable
Referenced by:: §8.14
Permalink:: http://dlmf.nist.gov/13.10.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §13.10(ii), §13.10 and Ch.13

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13.10.5

\int_{0}^{\infty}e^{-t}t^{b-1}{\mathbf{M}}\left(a,c,t\right)\,\mathrm{d}t=% \frac{\Gamma\left(b\right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a\right)% \Gamma\left(c-b\right)},

\Re\left(c-a\right)>\Re b>0

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $\Re$ : real part
Permalink:: http://dlmf.nist.gov/13.10.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §13.10(ii), §13.10 and Ch.13

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13.10.10

\int_{0}^{\infty}t^{\lambda-1}{\mathbf{M}}\left(a,b,-t\right)\,\mathrm{d}t=% \frac{\Gamma\left(\lambda\right)\Gamma\left(a-\lambda\right)}{\Gamma\left(a% \right)\Gamma\left(b-\lambda\right)},

0<\Re\lambda<\Re a

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\Re$ : real part
Keywords:: Mellin transform
Referenced by:: §8.14
Permalink:: http://dlmf.nist.gov/13.10.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §13.10(iii), §13.10 and Ch.13

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13.10.14

\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}{\mathbf{M}}\left(a,b,t\right)J_{\nu}% \left(2\sqrt{xt}\right)\,\mathrm{d}t=\frac{x^{\frac{1}{2}\nu}e^{-x}}{\Gamma% \left(b-a\right)}U\left(a,a-b+\nu+2,x\right),

x>0

-1<\Re\nu<2\Re\left(b-a\right)-\tfrac{1}{2}

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part and $x$ : real variable
Referenced by:: §13.10(v)
Permalink:: http://dlmf.nist.gov/13.10.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §13.10(v), §13.10 and Ch.13

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10: 13.4 Integral Representations

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13.4.1

{\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)\Gamma\left(b-a% \right)}\int_{0}^{1}e^{zt}t^{a-1}(1-t)^{b-a-1}\,\mathrm{d}t,

\Re b>\Re a>0

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part and $z$ : complex variable
A&S Ref:: 13.2.1 (in different form)
Referenced by:: §13.10(v), §13.29(iii)
Permalink:: http://dlmf.nist.gov/13.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §13.4(i), §13.4 and Ch.13

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13.4.2

{\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma\left(b-c\right)}\int_{0}^{1}{% \mathbf{M}}\left(a,c,zt\right)t^{c-1}(1-t)^{b-c-1}\,\mathrm{d}t,

\Re b>\Re c>0

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $z$ : complex variable and $c$ : arbitrary
Referenced by:: §13.10(vi)
Permalink:: http://dlmf.nist.gov/13.4.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §13.4(i), §13.4 and Ch.13

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13.4.3

{\mathbf{M}}\left(a,b,-z\right)=\frac{z^{\frac{1}{2}-\frac{1}{2}b}}{\Gamma% \left(a\right)}\int_{0}^{\infty}e^{-t}t^{a-\frac{1}{2}b-\frac{1}{2}}J_{b-1}% \left(2\sqrt{zt}\right)\,\mathrm{d}t,

\Re a>0

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part and $z$ : complex variable
Referenced by:: §13.10(v)
Permalink:: http://dlmf.nist.gov/13.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §13.4(i), §13.4 and Ch.13

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13.4.9

{\mathbf{M}}\left(a,b,z\right)=\frac{\Gamma\left(1+a-b\right)}{2\pi\mathrm{i}% \Gamma\left(a\right)}\int_{0}^{(1+)}e^{zt}t^{a-1}{(t-1)^{b-a-1}}\,\mathrm{d}t,

b-a\neq 1,2,3,\dots

\Re a>0

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.4.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §13.4(ii), §13.4 and Ch.13

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13.4.12

{\mathbf{M}}\left(a,c,z\right)=\frac{\Gamma\left(b\right)}{2\pi\mathrm{i}}z^{1% -b}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}{{}_{2}{\mathbf{F}}_{1}}\left(a,b;c;% \ifrac{1}{t}\right)\,\mathrm{d}t,

b\neq 0,-1,-2,\dots

\left|\operatorname{ph}z\right|<\frac{1}{2}\pi

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ or ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{% \mathbf{b}}};\NVar{z}\right)$ : scaled (or Olver’s) generalized hypergeometric function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $(\NVar{a},\NVar{b})$ : open interval, $\operatorname{ph}$ : phase and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.4.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §13.4(ii), §13.4 and Ch.13

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