15 Hypergeometric Function Properties 15.5 Derivatives and Contiguous Functions 15.7 Continued Fractions

§15.6 Integral Representations

ⓘ

Keywords:: Mellin–Barnes type, hypergeometric function, integral representations
A&S Ref:: 15.3
Notes:: See Andrews et al. (1999, §§2.2, 2.4, 2.9), Erdélyi et al. (1953a, §2.1.3), and Whittaker and Watson (1927, pp. 290–291). (15.6.3) follows from (15.6.4). (15.6.8) follows from (15.6.9) with $\lambda=a+b$ .
Referenced by:: §8.17(iii), Erratum (V1.0.14) for Equation (15.6.8), Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9), Erratum (V1.0.5) for Clarifications
Permalink:: http://dlmf.nist.gov/15.6
Addition (effective with 1.1.5):: The integral representation (15.6.2_5) was added.
Clarification (effective with 1.0.23):: The sentence immediately following (15.6.9), “These representations are valid when $|\operatorname{ph}\left(1-z\right)|<\pi$ , except (15.6.6) which holds for $|\operatorname{ph}\left(-z\right)|<\pi$ .”, has been removed. These formula constraints have been directly inserted into the constraints for Equations (15.6.1)–(15.6.9).
Clarification (effective with 1.0.14):: A sentence was added below (15.6.9) to point out that (15.6.8) could be rewritten as a fractional integral.
Clarification (effective with 1.0.5):: The description of the integration contour for (15.6.5), depicted in Figure 15.6.1, has been expanded.
See also:: Annotations for Ch.15

The function $\mathbf{F}\left(a,b;c;z\right)$ (not $F\left(a,b;c;z\right)$ ) has the following 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: TeX, pMML, png Addition (effective with 1.0.23): The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , 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: TeX, pMML, png Addition (effective with 1.0.23): The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , 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_5		$\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(b\right)\Gamma\left(c-b% \right)}\int_{0}^{\infty}\frac{t^{b-1}\left(t+1\right)^{a-c}}{\left(t-zt+1% \right)^{a}}\,\mathrm{d}t,$
$\|\operatorname{ph}\left(1-z\right)\|<\pi$ ; $\Re c>\Re b>0$ .
ⓘ 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: §15.6, Erratum (V1.1.5) for Additions Permalink: http://dlmf.nist.gov/15.6.E2_5 Encodings: TeX, pMML, png Addition (effective with 1.1.5): This equation was added. See also: Annotations for §15.6 and Ch.15

15.6.3		$\mathbf{F}\left(a,b;c;z\right)={\mathrm{e}}^{-b\pi\mathrm{i}}\frac{\Gamma\left% (1-b\right)}{2\pi\mathrm{i}\Gamma\left(c-b\right)}\int_{\infty}^{(0+)}\frac{t^% {b-1}(t+1)^{a-c}}{(t-zt+1)^{a}}\,\mathrm{d}t,$
$\|\operatorname{ph}\left(1-z\right)\|<\pi$ ; $b\neq 1,2,3,\dots$ , $\Re\left(c-b\right)>0$ .
ⓘ 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 $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : 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, §15.6 Permalink: http://dlmf.nist.gov/15.6.E3 Encodings: TeX, pMML, png Addition (effective with 1.0.23): The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , 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

15.6.4		$\mathbf{F}\left(a,b;c;z\right)={\mathrm{e}}^{-b\pi\mathrm{i}}\frac{\Gamma\left% (1-b\right)}{2\pi\mathrm{i}\Gamma\left(c-b\right)}\int_{1}^{(0+)}\frac{t^{b-1}% (1-t)^{c-b-1}}{(1-zt)^{a}}\,\mathrm{d}t,$
$\|\operatorname{ph}\left(1-z\right)\|<\pi$ ; $b\neq 1,2,3,\dots$ , $\Re\left(c-b\right)>0$ .
ⓘ 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 $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : 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, §15.6 Permalink: http://dlmf.nist.gov/15.6.E4 Encodings: TeX, pMML, png Addition (effective with 1.0.23): The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , 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

15.6.5		$\mathbf{F}\left(a,b;c;z\right)={\mathrm{e}}^{-c\pi\mathrm{i}}\Gamma\left(1-b% \right)\Gamma\left(1+b-c\right)\*\frac{1}{4\pi^{2}}\int_{A}^{(0+,1+,0-,1-)}% \frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\,\mathrm{d}t,$
$\|\operatorname{ph}\left(1-z\right)\|<\pi$ ; $b,c-b\neq 1,2,3,\dots$ .
ⓘ 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 $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : 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 Keywords: Mellin transform Referenced by: Figure 15.6.1, Figure 15.6.1, §15.6, §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9) Permalink: http://dlmf.nist.gov/15.6.E5 Encodings: TeX, pMML, png Addition (effective with 1.0.23): The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , 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: TeX, pMML, png Addition (effective with 1.0.23): The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , 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: TeX, pMML, png Addition (effective with 1.0.23): The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , 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: TeX, pMML, png Addition (effective with 1.0.23): The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , 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

15.6.9		$\mathbf{F}\left(a,b;c;z\right)=\int_{0}^{1}\frac{t^{d-1}(1-t)^{c-d-1}}{(1-zt)^% {a+b-\lambda}}\mathbf{F}\left({\lambda-a,\lambda-b\atop d};zt\right)\mathbf{F}% \left({a+b-\lambda,\lambda-d\atop c-d};\frac{(1-t)z}{1-zt}\right)\,\mathrm{d}t,$
$\|\operatorname{ph}\left(1-z\right)\|<\pi$ ; $\lambda\in\mathbb{C}$ , $\Re c>\Re d>0$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbb{C}$ : complex plane, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\int$ : 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 Notes: With $\lambda=a+b$ , this equation specializes to (15.6.8). Referenced by: (15.6.8), §15.6, §15.6, Erratum (V1.0.18) for Equation (15.6.9), 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.E9 Encodings: TeX, pMML, png Addition (effective with 1.0.23): The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , 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. Addition (effective with 1.0.18): The constraint $\lambda\in\mathbb{C}$ was added. See also: Annotations for §15.6 and Ch.15

In all cases the integrands are continuous functions of $t$ on the integration paths, except possibly at the endpoints. Note that (15.6.8) can be rewritten as a fractional integral. In addition:

In (15.6.1) all functions in the integrand assume their principal values.

In (15.6.2) the point $\ifrac{1}{z}$ lies outside the integration contour, $t^{b-1}$ and $(t-1)^{c-b-1}$ assume their principal values where the contour cuts the interval $(1,\infty)$ , and $(1-zt)^{a}=1$ at $t=0$ .

In (15.6.3) the point $\ifrac{1}{(z-1)}$ lies outside the integration contour, the contour cuts the real axis between $t=-1$ and $0$ , at which point $\operatorname{ph}t=\pi$ and $\operatorname{ph}\left(1+t\right)=0$ .

In (15.6.4) the point $\ifrac{1}{z}$ lies outside the integration contour, and at the point where the contour cuts the negative real axis $\operatorname{ph}t=\pi$ and $\operatorname{ph}\left(1-t\right)=0$ .

In (15.6.5) the integration contour starts and terminates at a point $A$ on the real axis between $0$ and $1$ . It encircles $t=0$ and $t=1$ once in the positive direction, and then once in the negative direction. See Figure 15.6.1. At the starting point $\operatorname{ph}t$ and $\operatorname{ph}\left(1-t\right)$ are zero. If desired, and as in Figure 5.12.3, the upper integration limit in (15.6.5) can be replaced by $(1+,0+,1-,0-)$ . However, this reverses the direction of the integration contour, and in consequence (15.6.5) would need to be multiplied by $-1$ .

In (15.6.6) the integration contour separates the poles of $\Gamma\left(a+t\right)$ and $\Gamma\left(b+t\right)$ from those of $\Gamma\left(-t\right)$ , and $(-z)^{t}$ has its principal value.

In (15.6.7) the integration contour separates the poles of $\Gamma\left(a+t\right)$ and $\Gamma\left(b+t\right)$ from those of $\Gamma\left(c-a-b-t\right)$ and $\Gamma\left(-t\right)$ , and $(1-z)^{t}$ has its principal value.

In each of (15.6.8) and (15.6.9) all functions in the integrand assume their principal values.

See accompanying text — Figure 15.6.1: $t$ -plane. Contour of integration in (15.6.5). Magnify

15.5 Derivatives and Contiguous Functions 15.7 Continued Fractions

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§15.6 Integral Representations

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