1550251515-Classical_Complex_Analysis__Gonzalez_

Singularities/Residues/ Applications 677

which gives an alternative method (often advantageous) for the evaluation
of an integral in terms of the residue of the integrand at infinity.

Example To evaluate Ic+(z^3 + 2)/(z^4 + 1) dz, where c+: z = 2eit, 0 ~
t ~ 271".
The integrand has simple poles at the points ak (k = 1,2,3,4), where
at = -1, i.e., at the zeros of z^4 + 1 = 0 (Fig. 9.7). Since all those poles lie
within C*, we may use either (9.9-1) or (9.9-4) in evaluating the integral.
First method. Applying (9.8-2) we have

z^3 + 2 ai + 2 1 1 1 1 1

Res --= --= - + - - = - - -ak

z=a1o z4 + 1 4ai 4 2 ai 4 2

Hence by (9.9-1),

--dz = 27ri """"' - - -ak

J

za + 2.

4

( 1 1 )

c+ z

4

+l f:i 4 2

= 27ri (1 - % t ak) = 27ri

k=l

since "E!=l ak = 0.

Second method. By using (9.7-4) we have

Hence by (9.9-4),

Fig. 9.7

Res za + 2 == -Res ~ 1 + 2z3 = -1

z=oo z^4 + 1 z=O z 1 + z^4

J

z: +

2

dz= -27ri(-1) = 27ri

z + 1

c+