1550251515-Classical_Complex_Analysis__Gonzalez_

666 Chapter 9

f. Thus oo is a pole of f iff the degree of the denominator is smaller than

the degree of the numerator. In any case, the rational function f = P/Q

is meromorphic in C* .with a finite number of poles equal to max(m, n).

Corollary 9.2 Any rational function can be decomposed into simple frac-

tions of the form A/(z - a)1' and a polynomial (which may reduce to a

constant).

Proof It follows at once from (9.6-1).

9. 7 Residues

Definition 9.11 Let z = a ~ oo be either a regular point or an isolated
singularity off' and let c+: ( -a = reit' 0 :::;; t :::;; 271"' be a circle contained

in a deleted neighborhood NHa) of a where f is analytic (Fig. 9.4). Then

the residue of f at a is defined by

~~f(z) = 2 ~i j f(()d( (9.7-1)

c+

As before, the notation c+ is used to emphasize that C is described once

in the positive direction.
The value obtained in (9. 7-1) is independent of the radius r of C as long
as C* remains in 0 < lz - al < 5.
If a is a regular point off, it is clear that Resz=a f(z) = 0 (Theo-
rem 7.21). However, we may have Resz=a f(z) = 0 at isolated singularities
of certain functions (see Example 2). If f(z) = I:~~-oo Am(z -a)m is the
Laurent expansion of f(z) valid in 0 < lz -aJ < 8, we have (8.18-2)

Am=~ J f(()d(

27l"i (( - a)m+l

c+

For m = -1 we obtain

A-1 = 2 ~i j f(()d( = ~~f(z) (9.7-2)

c+

Fig. 9.4