1550251515-Classical_Complex_Analysis__Gonzalez_

594 Chapter^8

Fig. 8.8

where r < R, then the function f(z) = f1(z) + fz(z) is said to be the
function defined by the Laurent series in the ring r < lz -al < R. Clearly,
this function is analytic in that ring.
Next, we wish to prove that, conversely, any analytic function in a ring
r < lz -al < R can be expanded into a Laurent series which converges to
the. function for every z in that ring.

8.18 The Laurent Series Expansion Theorem

Theorem 8.42 If f ( z) is analytic in the ring

G = {z: r < lz -al< R}

then for any z E G we have

+oo

f(z) = L Am(Z - ar

m=-oo

where

Am= _1 J f(t)dt

27ri (t-a)m+l

r

(8.18-1)

(8.18-2)

and r is any simple close contour contained in G and such that Dr(a) = +1.

Proof For any z such that r < lz -al = p < R, let

where 0 ~ (J ~ 271" and r' < r^1 < p < R' < R (Fig. 8.9). Then f(z) is

analytic in r' ~ lz -al ::::; R' and by Cauchy's formula we have

f(z) = ~ j f(t)dt _ ~ j f(t)dt

27rz t - z 2m t - z

(8.18-3)

C1 C2