1550251515-Classical_Complex_Analysis__Gonzalez_

Singularities/Residues/ Applications 699

the real axis is regarded as belonging to the upper half-plane. Formula
(9.11-21) applies, in particular, to rational functions f(z) = P(z)/Q(z),
with P and Q prime to each other and the degree of Q exceeding at least
by two that of P.

Example To evaluate (PV) J~ 00 dx/(x^3 - 1). Consider f c+ dz/(z^3 - 1),

where c+ is the same contour as in Fig. 9.16. In this case the integrand

has a simple pole p = 1 on the real axis, and a simple pole b = ei^2 n:/^3 in
the upper half-plane. Since

and

Resf(z) = __!._ = ~ei^2 n:/^3

z=b 3b^2 3

1

~~ff(z)= 3

by applying (9.11-20) we get

(PV) loo __±:___ = 27ri ( ~ ei2n:/a + ~)

_ 00 x^3 -1 3 6

= -71"-J3

3

To evaluate (PV) f 0

00

dx / ( x^3 -1) we consider the integral J a+ dz/ ( z^3 -1),

where c+ is now the boundary of the sector OAB with central angle 271" /3

and indentations at the poles z = 1 and z = b = ei^2 n:/^3 located, respectively,
on the initial and terminal rays of the sector (Fig. 9.17). Since there are

no singularities of the integrand inside c+' we have

rl-r ±:__ + J :!:! + {R ±: + J __:!:!

Jo x^3 - 1 z^3 - 1 }Hr x^3 - 1 z^3 - 1

-Yi" r+

l

+ b+r'b --dz + J --dz + 1° --dz = 0 (9.11-22)

Rb z3 - 1 z3 - 1 b-r'b z3 - 1

'Y2

As before,

1. im j --dz = - -3 1. 7ri

r->O z^3 - 1

'Yt

1 1m. j --= dz

r^1 ->0 z^3 - 1