1550251515-Classical_Complex_Analysis__Gonzalez_

608 Chapter^8

where () ;:::: Arg w and k is any integer. Then

1 1 ()

• 2
  . logw = -
  2
  . lnlwl + -
  2

+k

1i'Z 1i'Z 1i'
and it follows that

F(w) = f ( 2 ~i ln JwJ + ~ + k) = f ( 2 ~i ln JwJ + 2 ~)

since the integer le is a period of f. Hence F( w) does not depend on the

value chosen for arg w.
Also, from F(w) = f(z) we get

F'(w) = J'(z) dz = f'(z) = f'(~)

dw dw/dz 21l'iw

so that F( w) is analytic in r 2 < Jw I < ri. It should be noted that in a

neighborhood of any point Wo such that r2 < Jwo I < ri we may write

w

logw =log - + logwo

Wo

which defines a single-valued analytic function in that neighborhood, in
fact, in the region Jw/w 0 J + w/w 0 '/= 0, i.e., in the plane with the ray

w = -tw 0 (t 2: 0) removed (Fig. 8.14).

Example If f(z) = cos27i'z =^1 / 2 (e^2 n:iz + e-^2 n:iz) we have

By applying Laurent expansion to F( w ), we get

+oo

F(w) = L Anwn

n=-oo

Fig. 8.Jl4

(8.19-6)