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5 Gamma FunctionProperties

§5.12 Beta Function

In this section all fractional powers have their principal values, except where noted otherwise. In (5.12.1)–(5.12.4) it is assumed a>0 and b>0.

Euler’s Beta Integral

5.12.1 B(a,b)=01ta1(1t)b1dt=Γ(a)Γ(b)Γ(a+b).
5.12.2 0π/2sin2a1θcos2b1θdθ=12B(a,b).
5.12.3 0ta1dt(1+t)a+b=B(a,b).
5.12.4 01ta1(1t)b1(t+z)a+bdt=B(a,b)(1+z)azb,
5.12.5 0π/2(cost)a1cos(bt)dt=π2a1aB(12(a+b+1),12(ab+1)),
5.12.6 0π(sint)a1eibtdt=π2a1eiπb/2aB(12(a+b+1),12(ab+1)),
5.12.7 0cosh(2bt)(cosht)2adt=4a1B(a+b,ab),
5.12.8 12πdt(w+it)a(zit)b=(w+z)1ab(a+b1)B(a,b),
(a+b)>1, w>0, z>0.

In (5.12.8) the fractional powers have their principal values when w>0 and z>0, and are continued via continuity.

5.12.9 12πicic+ita(1t)1bdt=1bB(a,b),
0<c<1, (a+b)>0.
5.12.10 12πi0(1+)ta1(t1)b1dt=sin(πb)πB(a,b),

with the contour as shown in Figure 5.12.1.

See accompanying text
Figure 5.12.1: t-plane. Contour for first loop integral for the beta function. Magnify

In (5.12.11) and (5.12.12) the fractional powers are continuous on the integration paths and take their principal values at the beginning.

5.12.11 1e2πia1(0+)ta1(1+t)abdt=B(a,b),

when b>0, a is not an integer and the contour cuts the real axis between 1 and the origin. See Figure 5.12.2.

See accompanying text
Figure 5.12.2: t-plane. Contour for second loop integral for the beta function. Magnify

Pochhammer’s Integral

When a,b

5.12.12 P(1+,0+,1,0)ta1(1t)b1dt=4eπi(a+b)sin(πa)sin(πb)B(a,b),

where the contour starts from an arbitrary point P in the interval (0,1), circles 1 and then 0 in the positive sense, circles 1 and then 0 in the negative sense, and returns to P. It can always be deformed into the contour shown in Figure 5.12.3.

See accompanying text
Figure 5.12.3: t-plane. Contour for Pochhammer’s integral. Magnify