1550251515-Classical_Complex_Analysis__Gonzalez_

Singularities/Residues/ Applications 667

Thus the residue of f at a is equal to the coefficient A_ 1 in the Laurent

expansion of f(z) valid in a deleted neighborhood of a.t

Examples

1. Resz=O ~ = Resz=o( * + ~ + ~ + · · ·) = 1.


2. Resz=l (z!l) 2 = 2 ~i fa+ ((~}) 2 = 0, by 7.8-13.

Alternatively, we may observe that 1/(z - 1)^2 is just the Laurent ~x­

pansion of the same function in the vicinity of z = 1. Hence A_ 1 = 0 and

the residue at z = 1 is zero.

Definition 9.12 ;Let a = oo be either a regular point or an isolated sin-
gularity off, and let c-: (=Re-it, 0:::; t:::; 27r, be a circle contained in

a deleted neighborhood N' ( oo) of oo where f is analytic. Then the residue

of f at oo is defined by

~~f(z) = 2 ~i j J(()d( (9.7-3)

a-

The notation c-is used to emphasize that in this case the circle. C
is described once in the negative (clockwise) direction. Again, the value

obtained in (9.7-3) is independent of Ras long as C* remains in a region

Ro < JzJ < oo where f is analytic.

Let ( = 1/z, d( = -dz/z^2 • If ( = Re-it we have z = (1/R)eit

(0:::; t:::; 27r), so that c-maps into the circle 1+: z = 1/Reit, which is de-
scribed once in the positive direction, as the notation indicated (Fig. 9.5).
Then (9.7-3) becomes

Resf(z)=--^1 j dz 1 ( 1)

2

. f(l/z)2 =-Res2f -
z=oo 7l"Z Z z=O Z Z
(9.7-4)
-y+

If f(z) = E~: Bmzm is the Laurent expansion of f(z) valid for JzJ >

R 0 , and we take R > R 0 , we get

B = _1 J f(()d(

m 27l"i (m+l

a+

tThis fact was used by A. L. Cauchy to define the residue of a function at a pole
(Sur un nouveau genre de calcul analogue au calcul infinitesimal, Exer. Math.,
Paris, 1826).