1550251515-Classical_Complex_Analysis__Gonzalez_

668 Chapter9

R

Fig. 9.5

so that

B_1 = 2 ~i J J(() d(

c+

Hence we obtain

Res J(z) = -

2

1

z=oo 7rZ. J J(() d( = -B-1 (9.7-5)

c-

i.e., the residue if f at oo is equal to the negative of the coefficient B- 1 in

the Laurent expansion of J(z) valid in a deleted neighborhood of oo.

Example Let f(z) = z/(z + 2). We have by using (9.7-4),

1 1 1 2

Res f(z) = - Res - --= - Res -(1 - 2z + 4z - · · ·)

z=oo z=O z2 1 + 2z z=O z^2

= - Res z=O ( 2_ z 2 - ~ z + 4 - · · ·) = +2

Alternatively, since

1 2 4

f(z)= 1+2/z =1--;; + z2 -··· for JzJ >^2

by applying (9. 7-5) we obtain Resz=oo J(z) = +2. We note that oo is a

regular point of f(z) = z/(z + 2). Hence the residue at oo of a function

that is regular at this point may not be equal to zero.

Note The residue of a function at a nonisolated sihgularity is not defined.