§15.6 Integral Representations

The function $\mathbf{F}(a,b;c;z)$ (not $F(a,b;c;z)$) has the following integral representations:

15.6.1		$$\mathbf{F}(a,b;c;z)=\frac{1}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-b\right)}{\int }_0^1\frac{t^{b-1}{(1-t)}^{c-b-1}}{{(1-zt)}^a}dt,$$
; $\mathrm{\Re }c>\mathrm{\Re }b>0$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ 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 $\mathbf{F}(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula. See also: Annotations for §15.6 and Ch.15

15.6.2		$$\mathbf{F}(a,b;c;z)=\frac{\mathrm{\Gamma }\left(1+b-c\right)}{2\pi \mathrm{i}\mathrm{\Gamma }\left(b\right)}{\int }_0^{(1+)}\frac{t^{b-1}{(t-1)}^{c-b-1}}{{(1-zt)}^a}dt,$$
; $c-b\ne 1,2,3,\mathrm{\dots }$, $\mathrm{\Re }b>0$.
ⓘ 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, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ 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 $\mathbf{F}(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula. See also: Annotations for §15.6 and Ch.15

15.6.3		$$\mathbf{F}(a,b;c;z)={\mathrm{e}}^{-b\pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(1-b\right)}{2\pi \mathrm{i}\mathrm{\Gamma }\left(c-b\right)}{\int }_{\mathrm{\infty }}^{(0+)}\frac{t^{b-1}{(t+1)}^{a-c}}{{(t-zt+1)}^a}dt,$$
; $b\ne 1,2,3,\mathrm{\dots }$, $\mathrm{\Re }\left(c-b\right)>0$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ 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 $\mathbf{F}(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula. See also: Annotations for §15.6 and Ch.15

15.6.4		$$\mathbf{F}(a,b;c;z)={\mathrm{e}}^{-b\pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(1-b\right)}{2\pi \mathrm{i}\mathrm{\Gamma }\left(c-b\right)}{\int }_1^{(0+)}\frac{t^{b-1}{(1-t)}^{c-b-1}}{{(1-zt)}^a}dt,$$
; $b\ne 1,2,3,\mathrm{\dots }$, $\mathrm{\Re }\left(c-b\right)>0$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ 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 $\mathbf{F}(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula. See also: Annotations for §15.6 and Ch.15

15.6.5		$$\mathbf{F}(a,b;c;z)=\begin{array}{l}{\mathrm{e}}^{-c\pi \mathrm{i}}\mathrm{\Gamma }\left(1-b\right)\mathrm{\Gamma }\left(1+b-c\right)\frac{1}{4{\pi }^2}\\ \times {\int }_A^{(0+,1+,0-,1-)}\frac{t^{b-1}{(1-t)}^{c-b-1}}{{(1-zt)}^a}dt,\end{array}$$
; $b,c-b\ne 1,2,3,\mathrm{\dots }$.
ⓘ 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, $\mathrm{ph}$: phase, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ 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 $\mathbf{F}(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula. See also: Annotations for §15.6 and Ch.15

15.6.6		$$\mathbf{F}(a,b;c;z)=\frac{1}{2\pi \mathrm{i}\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a+t\right)\mathrm{\Gamma }\left(b+t\right)\mathrm{\Gamma }\left(-t\right)}{\mathrm{\Gamma }\left(c+t\right)}{(-z)}^{t}dt,$$
; $a,b\ne 0,-1,-2,\mathrm{\dots }$.
ⓘ 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, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ 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 $\mathbf{F}(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula. See also: Annotations for §15.6 and Ch.15

15.6.7		$$\mathbf{F}(a,b;c;z)=\begin{array}{l}\frac{1}{2\pi \mathrm{i}\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)}\\ \times {\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\begin{array}{l}\mathrm{\Gamma }\left(a+t\right)\mathrm{\Gamma }\left(b+t\right)\mathrm{\Gamma }\left(c-a-b-t\right)\\ \times \mathrm{\Gamma }\left(-t\right){(1-z)}^{t}dt,\end{array}\end{array}$$
; $a,b,c-a,c-b\ne 0,-1,-2,\mathrm{\dots }$.
ⓘ 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, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ 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 $\mathbf{F}(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula. See also: Annotations for §15.6 and Ch.15

15.6.8		$$\mathbf{F}(a,b;c;z)=\frac{1}{\mathrm{\Gamma }\left(c-d\right)}{\int }_0^1\mathbf{F}(a,b;d;zt)t^{d-1}{(1-t)}^{c-d-1}dt,$$
; $\mathrm{\Re }c>\mathrm{\Re }d>0$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ 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 $\mathbf{F}(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula. See also: Annotations for §15.6 and Ch.15

15.6.9		$$\mathbf{F}(a,b;c;z)={\int }_0^1\frac{t^{d-1}{(1-t)}^{c-d-1}}{{(1-zt)}^{a+b-\lambda }}\mathbf{F}(\genfrac{}{}{0pt}{}{\lambda -a,\lambda -b}{d};zt)\mathbf{F}(\genfrac{}{}{0pt}{}{a+b-\lambda ,\lambda -d}{c-d};\frac{(1-t)z}{1-zt})dt,$$
; $\lambda \in ℂ$, $\mathrm{\Re }c>\mathrm{\Re }d>0$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $ℂ$: complex plane, $dx$: differential of $x$, $\in$: element of, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ 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 $\mathbf{F}(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula. Addition (effective with 1.0.18): The constraint $\lambda \in ℂ$ 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 $1/z$ lies outside the integration contour, $t^{b-1}$ and ${(t-1)}^{c-b-1}$ assume their principal values where the contour cuts the interval $(1,\mathrm{\infty })$, and ${(1-zt)}^a=1$ at $t=0$.

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

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

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 $\mathrm{ph}t$ and $\mathrm{ph}\left(1-t\right)$ 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 $\mathrm{\Gamma }\left(a+t\right)$ and $\mathrm{\Gamma }\left(b+t\right)$ from those of $\mathrm{\Gamma }\left(-t\right)$, and ${(-z)}^t$ has its principal value.

In (15.6.7) the integration contour separates the poles of $\mathrm{\Gamma }\left(a+t\right)$ and $\mathrm{\Gamma }\left(b+t\right)$ from those of $\mathrm{\Gamma }\left(c-a-b-t\right)$ and $\mathrm{\Gamma }\left(-t\right)$, 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.