# §15.6 Integral Representations

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), §15.6, §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
 15.6.3 $\mathbf{F}\left(a,b;c;z\right)=e^{-b\pi\mathrm{i}}\frac{\Gamma\left(1-b\right)% }{2\pi\mathrm{i}\Gamma\left(c-b\right)}\int_{\infty}^{(0+)}\frac{t^{b-1}(t+1)^% {a-c}}{(t-zt+1)^{a}}\mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $b\neq 1,2,3,\dots$, $\Re\left(c-b\right)>0$. ⓘ 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 $\mathbf{F}\left(a,b;c;z\right)=e^{-b\pi\mathrm{i}}\frac{\Gamma\left(1-b\right)% }{2\pi\mathrm{i}\Gamma\left(c-b\right)}\int_{1}^{(0+)}\frac{t^{b-1}(1-t)^{c-b-% 1}}{(1-zt)^{a}}\mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $b\neq 1,2,3,\dots$, $\Re\left(c-b\right)>0$. ⓘ 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 $\mathbf{F}\left(a,b;c;z\right)=e^{-c\pi\mathrm{i}}\Gamma\left(1-b\right)\Gamma% \left(1+b-c\right)\*\frac{1}{4\pi^{2}}\int_{A}^{(0+,1+,0-,1-)}\frac{t^{b-1}(1-% t)^{c-b-1}}{(1-zt)^{a}}\mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $b,c-b\neq 1,2,3,\dots$. ⓘ 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 Source: Follows from (15.6.9) with $\lambda=a+b$. Referenced by: (15.6.9), §15.6, §15.6, §15.6, 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.