1550251515-Classical_Complex_Analysis__Gonzalez_

700 Chapter^9

Rb

b+r'b, --\ 'Y2 ~ f+

\

b~ \

\

b-r'b

~ 'Y1~

A

0 x

Fig. 9.JL7

r J dz 0

R::Uoo z^3 - 1 =

r+

lim ( {1-r _.!!::: + lR .!!:::_) - (PV) {R dx
r->O } 0 x^3 - 1 l+r x^3 - 1 - } 0 x^3 - 1
By using the equation of the ray OB, namely, z = xb, 0 :::; x :::; R, the
fifth and the last integrals in (9.11-22) combine to yield

b ( {Hr' ___.!!:::_ + lo ___.!!:::_) ---t -b (PV) {R dx

JR x3 - 1 1-r' x3 - 1 Jo x3 - 1

as r^1 ---+ 0. Hence taking limits in (9.11-22) as r ---+ 0, r^1 ---+ 0 and R---+ oo,

successively, we get

(1 - ei211"/3)(PV) 1"° ___.!!:::_ = 7f'i (1 + ei211"/3)

0 x^3 -1^3

or

[^00 dx 7f'i 1 + ei^2 11"/^3

(PV) Jo xa - 1 = 3 1 - ei211"/3

7f'

• 3v'3

Type V. (PV) J::' 00 f ( x) cos ax dx and (PV) J::' 00 f( x) sin ax dx, where

a > 0 and f ( x) satisfies the following conditions:

1. f ( x) has an extension f ( z) that is meromorphic in c (or at least on
   either Imz 2: 0 or Imz ~ 0).


2. f(z) has simple poles on the real axis.


3. f(z) = 0(1/z°'), a 2: 1, as z ---+ oo.

rt suffices to consider the integral f c+ eiaz JC z) dz, where c+ is a contour

of the type shown in Fig. 9.16, except that each pole p 8 of f(z) of the real