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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 F(a,b;c;z)=1Γ(b)Γ(cb)01tb1(1t)cb1(1zt)adt,
|ph(1z)|<π; c>b>0.
15.6.2 F(a,b;c;z)=Γ(1+bc)2πiΓ(b)0(1+)tb1(t1)cb1(1zt)adt,
|ph(1z)|<π; cb1,2,3,, b>0.
15.6.2_5 F(a,b;c;z)=1Γ(b)Γ(cb)0tb1(t+1)ac(tzt+1)adt,
|ph(1z)|<π; c>b>0.
15.6.3 F(a,b;c;z)=ebπiΓ(1b)2πiΓ(cb)(0+)tb1(t+1)ac(tzt+1)adt,
|ph(1z)|<π; b1,2,3,, (cb)>0.
15.6.4 F(a,b;c;z)=ebπiΓ(1b)2πiΓ(cb)1(0+)tb1(1t)cb1(1zt)adt,
|ph(1z)|<π; b1,2,3,, (cb)>0.
15.6.5 F(a,b;c;z)=ecπiΓ(1b)Γ(1+bc)14π2×A(0+,1+,0,1)tb1(1t)cb1(1zt)adt,
|ph(1z)|<π; b,cb1,2,3,.
15.6.6 F(a,b;c;z)=12πiΓ(a)Γ(b)iiΓ(a+t)Γ(b+t)Γ(t)Γ(c+t)(z)tdt,
|ph(z)|<π; a,b0,1,2,.
15.6.7 F(a,b;c;z)=12πiΓ(a)Γ(b)Γ(ca)Γ(cb)×iiΓ(a+t)Γ(b+t)Γ(cabt)×Γ(t)(1z)tdt,
|ph(1z)|<π; a,b,ca,cb0,1,2,.
15.6.8 F(a,b;c;z)=1Γ(cd)01F(a,b;d;zt)td1(1t)cd1dt,
|ph(1z)|<π; c>d>0.
15.6.9 F(a,b;c;z)=01td1(1t)cd1(1zt)a+bλF(λa,λbd;zt)F(a+bλ,λdcd;(1t)z1zt)dt,
|ph(1z)|<π; λ, c>d>0.

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, tb1 and (t1)cb1 assume their principal values where the contour cuts the interval (1,), and (1zt)a=1 at t=0.

In (15.6.3) the point 1/(z1) 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(1t)=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(1t) 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(cabt) and Γ(t), and (1z)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