1550251515-Classical_Complex_Analysis__Gonzalez_

732 Chapter9

(^0) -- R x
Fig. 9.30
24 [1 _ dx = 2rr
· Jo ·{/x^2 (1-x) V'3

25. f

1

{/x^2 (1-x) dx =

2

:,

Jo 9v3

26.1= .xa-l cos bx dx = ri:) cos ~arr

1= xa-lsinbxdx = ri:) sin ~arr, 0 <a< 1, b > 0

[Hint: Integrate f(z) = za-le-bz along the contour shown in Fig. 9.30.]

27 .. r= sinxa dx = r{l/a) cos~ (~)
} 0 xa a -1 2 a

-- x=----sm- - <a<l

1

= cosxa d r{l/a). 1 ( rr) 1 /

o xa a -1 2 a '^2

(Hint: Let xa = y and use Problem 26 with b = 1, followed by a

change of parameter.)

9.12 Summation of Certain Series by Using the Residue Theorem

The residue theorem may also be applied to derive the following proposi-
tion, which gives a method for' the summation of certain series.

Theorem 9.13 Suppose:

1. f { z) is a meromorphic function having just simple poles at the points

Zn= a+ n (n = 0, ±1, ±2, ... ) with residues f3n at those poles.

2. g( z) is analytic in some region G C C except for isolated singularities
   and such that zg(z)-> 0 as z-> oo. In particular, it may be that g(z)
   is a rational function P(z)/Q(z) with P and Q prime to each other
   and the degree of Q at least two units greater than that of P.