1550251515-Classical_Complex_Analysis__Gonzalez_

676 Chapter9

tyt

a, rc;;f~ a2

/ /

/

QC~ /'

an c+

Fig. 9.6

which is equivalent to (9.9-1).

Remark In view of Theorem 7.13 (strong form of the Cauchy-Goursat

theorem) the hypothesis in Theorem 9.9 can be weakened by requiring only

continuity of f(z) on c+. We shall refer to this as the strong form of the

residue theorem.

Corollary 9.3 If f has a finite number of isolated singularities in CC*, then

the sum of the residues of f at the finite singularities plus the residue at

oo (which may or may not be singular) is zero.

Proof Let c+: z = Reit, 0:::; t:::; 2?T, be a circle about the origin with ra-

dius R large enough so as to contain in its interior all the finite singularities

ak off (k = l, .. .,n). By Definition 9.12 we have

~~f(z) = 2 ~i j f(z)dz

c-

Hence

- Res f(z) = -.^1 J J(z) dz='""" n Res f(z)

z=oo 2?Ti L..J z=ak

a+ k=I

according to (9.9-2). Thus it follows that

n

L Res f(z) +Res f(z) = 0

k=l z=ak z=oo

(9.9-3)

Note If all the finite singularities of f lie within the contour C, then

from (9.9-1) and (9.9-3) we obtain

j f(z)dz=-~?Ti~~f(i) (9.9-4)

c+