1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 607

y

Fig. 8.l;l

we have

w = e2n:iz = -2n:y+2n:xi

so that

Hence if

(8.19-4)

we obtain

or

(8.19-5)

Clearly, if w is such that (8.19-5) is satisfied, then (8.19-4) holds, and

the points z = x+iy with x = (1/27r) argw, y = -(1/27r)ln lwl, are inverse

images of wall of which lie in the strip b 1 < y < b2. From (8.19-3) we have

Now let

1

z= -logw

27l"i

f(z) = f (~ logw) = F(w)

27ri

Although log w is multiple-valued, the periodicity of f compensates the

multiplicity of log w, and F( w) is a single-valued analytic function in the

ring r2 < lwl < r1. To see this, let

logw = ln lwl + i(B + 2k7r)