§15.6 Integral Representations

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).
A&S Ref:: 15.3
Keywords:: hypergeometric function
Referenced by:: §8.17(iii), Version 1.0.5 (October 1, 2012)
Permalink:: http://dlmf.nist.gov/15.6
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.
Reported 2012-09-19

The function $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ (not $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ ) has the following integral representations:

15.6.1 $\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(b\right)\mathop{\Gamma\/}\nolimits\!\left(c-b\right)}\int _{0}^{1}\frac{t^{{b-1}}(1-t)^{{c-b-1}}}{(1-zt)^{a}}dt,$ $\realpart{c}>\realpart{b}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.19(iii), §15.6
Permalink:: http://dlmf.nist.gov/15.6.E1
Encodings:: TeX, pMML, png

15.6.2 $\frac{\mathop{\Gamma\/}\nolimits\!\left(1+b-c\right)}{2\pi i\mathop{\Gamma\/}\nolimits\!\left(b\right)}\int _{0}^{{(1+)}}\frac{t^{{b-1}}(t-1)^{{c-b-1}}}{(1-zt)^{a}}dt,$ $c-b\neq 1,2,3,\dots$ , $\realpart{b}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.6
Permalink:: http://dlmf.nist.gov/15.6.E2
Encodings:: TeX, pMML, png

15.6.3 $e^{{-b\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(1-b\right)}{2\pi i\mathop{\Gamma\/}\nolimits\!\left(c-b\right)}\int _{{\infty}}^{{(0+)}}\frac{t^{{b-1}}(t+1)^{{a-c}}}{(t-zt+1)^{a}}dt,$ $b\neq 1,2,3,\dots$ , $\realpart{(c-b)}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.6, §15.6
Permalink:: http://dlmf.nist.gov/15.6.E3
Encodings:: TeX, pMML, png

15.6.4 $e^{{-b\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(1-b\right)}{2\pi i\mathop{\Gamma\/}\nolimits\!\left(c-b\right)}\int _{1}^{{(0+)}}\frac{t^{{b-1}}(1-t)^{{c-b-1}}}{(1-zt)^{a}}dt,$ $b\neq 1,2,3,\dots$ , $\realpart{(c-b)}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.6, §15.6
Permalink:: http://dlmf.nist.gov/15.6.E4
Encodings:: TeX, pMML, png

15.6.5 $e^{{-c\pi i}}\mathop{\Gamma\/}\nolimits\!\left(1-b\right)\mathop{\Gamma\/}\nolimits\!\left(1+b-c\right)\*\frac{1}{4\pi^{2}}\int _{A}^{{(0+,1+,0-,1-)}}\frac{t^{{b-1}}(1-t)^{{c-b-1}}}{(1-zt)^{a}}dt,$ $b,c-b\neq 1,2,3,\dots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: Figure 15.6.1, Figure 15.6.1, §15.6, §15.6
Permalink:: http://dlmf.nist.gov/15.6.E5
Encodings:: TeX, pMML, png

15.6.6 $\frac{1}{2\pi i\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(b\right)}\int _{{-i\infty}}^{{i\infty}}\frac{\mathop{\Gamma\/}\nolimits\!\left(a+t\right)\mathop{\Gamma\/}\nolimits\!\left(b+t\right)\mathop{\Gamma\/}\nolimits\!\left(-t\right)}{\mathop{\Gamma\/}\nolimits\!\left(c+t\right)}(-z)^{t}dt,$ $a,b\neq 0,-1,-2,\dots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $z$ : 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
Permalink:: http://dlmf.nist.gov/15.6.E6
Encodings:: TeX, pMML, png

15.6.7 $\frac{1}{2\pi i\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(b\right)\mathop{\Gamma\/}\nolimits\!\left(c-a\right)\mathop{\Gamma\/}\nolimits\!\left(c-b\right)}\int _{{-i\infty}}^{{i\infty}}\mathop{\Gamma\/}\nolimits\!\left(a+t\right)\mathop{\Gamma\/}\nolimits\!\left(b+t\right)\mathop{\Gamma\/}\nolimits\!\left(c-a-b-t\right)\mathop{\Gamma\/}\nolimits\!\left(-t\right)(1-z)^{t}dt,$ $a,b,c-a,c-b\neq 0,-1,-2,\dots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.6
Permalink:: http://dlmf.nist.gov/15.6.E7
Encodings:: TeX, pMML, png

15.6.8 $\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(c-d\right)}\int _{0}^{1}\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;d;zt\right)t^{{d-1}}(1-t)^{{c-d-1}}dt,$ $\realpart{c}>\realpart{d}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : 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
Permalink:: http://dlmf.nist.gov/15.6.E8
Encodings:: TeX, pMML, png

15.6.9 $\int _{0}^{1}\frac{t^{{d-1}}(1-t)^{{c-d-1}}}{(1-zt)^{{a+b-\lambda}}}\mathop{\mathbf{F}\/}\nolimits\!\left({\lambda-a,\lambda-b\atop d};zt\right)\mathop{\mathbf{F}\/}\nolimits\!\left({a+b-\lambda,\lambda-d\atop c-d};\frac{(1-t)z}{1-zt}\right)dt,$ $\realpart{c}>\realpart{d}>0$ .

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : 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
Permalink:: http://dlmf.nist.gov/15.6.E9
Encodings:: TeX, pMML, png

These representations are valid when $|\mathop{\mathrm{ph}\/}\nolimits\!\left(1-z\right)|<\pi$ , except (15.6.6) which holds for $|\mathop{\mathrm{ph}\/}\nolimits\!\left(-z\right)|<\pi$ . In all cases the integrands are continuous functions of $t$ on the integration paths, except possibly at the endpoints. 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 $\mathop{\mathrm{ph}\/}\nolimits t=\pi$ and $\mathop{\mathrm{ph}\/}\nolimits\!\left(1+t\right)=0$ .

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 $\mathop{\mathrm{ph}\/}\nolimits t=\pi$ and $\mathop{\mathrm{ph}\/}\nolimits\!\left(1-t\right)=0$ .

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

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

Figure 15.6.1: $t$ -plane. Contour of integration in (15.6.5).

Keywords:: Pochhammer double-loop contour
Referenced by:: §15.6, §15.6
Permalink:: http://dlmf.nist.gov/15.6.F1
Encodings:: pdf, png