§15.6 Integral Representations

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

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.