1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 593

A Laurent series is often written in the form

+oo

'""" ~ A m ( z - a )m = · .. + A_n + · · · + --A-1 + Ao +Ai ( z - a )

m=-oo (z - a)n z - a

Also, we write

+oo oo oo

L IAm(z - a)ml = L IA-n(z -a)-nl + L IAn(z - atl

m=-oo n=l n=O

and say that the Laurent series converges absolutely for some z iff each

series on the right-hand side converges for that value of z.

8.17 Region of Convergence

Let

1 _· /iAI

R

= n-+OO limvlAnl and '(' = n-+oo lira v'IA-n I

and suppose that R > O, 'T' < oo. Then the series Z:::'=o An(z - a)n con-

verges absolutely for every z such that lz-al R,

while the series z=:, 1 A_n(z - a)-n converges absolutely for every z such

that lz - al > 'T' and diverges if lz - al < 'T'. Hence if r < R, there is

a common region of absolute convergence, namely, the ring or annulus

'T' < lz -al < R (Fig. 8.8). This is the region of convergence of the Laurent

series. If 'T' = R, the series may converge only at a subset of points of the

circle lz - al = 'T' = R. If 'T' > R, the series diverges everywhere. It follows

fiom Theorems 4.19 and 8.41 that if r < R, the Laurent series converges

uniformly in any smaller ring

,,. < r' :::; lz - al :::; R' < R·

as well as in any compact subset of r < lz - al < R.

If we let

00

fi(z) = L An(z -at for lz - al< R

n=O

and

00

fz(z) = L A-n(z -a)-n for lz - al> r

n=l