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15 Hypergeometric FunctionProperties

§15.6 Integral Representations

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

15.6.1 1Γ(b)Γ(c-b)01tb-1(1-t)c-b-1(1-zt)adt,
c>b>0.
15.6.2 Γ(1+b-c)2πiΓ(b)0(1+)tb-1(t-1)c-b-1(1-zt)adt,
c-b1,2,3,, b>0.
15.6.3 e-bπiΓ(1-b)2πiΓ(c-b)(0+)tb-1(t+1)a-c(t-zt+1)adt,
b1,2,3,, (c-b)>0.
15.6.4 e-bπiΓ(1-b)2πiΓ(c-b)1(0+)tb-1(1-t)c-b-1(1-zt)adt,
b1,2,3,, (c-b)>0.
15.6.5 e-cπiΓ(1-b)Γ(1+b-c)14π2A(0+,1+,0-,1-)tb-1(1-t)c-b-1(1-zt)adt,
b,c-b1,2,3,.
15.6.6 12πiΓ(a)Γ(b)-iiΓ(a+t)Γ(b+t)Γ(-t)Γ(c+t)(-z)tdt,
a,b0,-1,-2,.
15.6.7 12πiΓ(a)Γ(b)Γ(c-a)Γ(c-b)-iiΓ(a+t)Γ(b+t)Γ(c-a-b-t)Γ(-t)(1-z)tdt,
a,b,c-a,c-b0,-1,-2,.
15.6.8 1Γ(c-d)01F(a,b;d;zt)td-1(1-t)c-d-1dt,
c>d>0.
15.6.9 01td-1(1-t)c-d-1(1-zt)a+b-λF(λ-a,λ-bd;zt)F(a+b-λ,λ-dc-d;(1-t)z1-zt)dt,
c>d>0.

These representations are valid when |ph(1-z)|<π, except (15.6.6) which holds for |ph(-z)|<π. 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 1/z lies outside the integration contour, tb-1 and (t-1)c-b-1 assume their principal values where the contour cuts the interval (1,), 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 pht=π and ph(1+t)=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 pht=π and ph(1-t)=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 pht and ph(1-t) 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 Γ(a+t) and Γ(b+t) from those of Γ(-t), and (-z)t has its principal value.

In (15.6.7) the integration contour separates the poles of Γ(a+t) and Γ(b+t) from those of Γ(c-a-b-t) and Γ(-t), 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