1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 609

which is absolutely convergent in the ring r 2 < lwl < ri and uniformly

convergent in r2 < r~ ~ lwl ~ ri < ri. Passing back to the variable z,

we obtain

+oo

J(z) = I:: Ane211"inz (8.19-7)

n=-oo

which converges absolutely in the strip bi < Imz < b2, and uniformly

on ci ~ Im z ~ c 2 , where e-^2 11"c1 = ri, e-^2 11"c2 = r~. The coefficient An

in (8.19-6) is given by

An= 2 ~i J ~~~~ dw (8.19-8)

r

where r: w = rei^8 , r2 < r < ri, 0 ~ () ~ 27r.

To express (8.19-8) in terms of z we note that the line segment z = x+ib,
where 0 ~ x ~ 1 and b is a constant such that bi < b < b2, is transformed
by (8.19-3) into the circle

contained in the ring r2 < Jwl < ri, and it suffices to chose b so that

e-^2 11"b = r in order to haver. We note also that 27rx = fJ = Argw, since
0 ~ x < 1 implies that 0 ~ () < 27r. Thus the line segment z = x + ib,

0 ~ X ~ 1, e-^2 11"b = r, is mapped by W = e^2 11"iZ into the Circler described

once in the positive direction (Fig. 8.15). From

w = reie = re211"ix = e211"i(x+ib)

we get

dw = re^2 11"ix • 27ri dx

v

b z = x + ib

0 x

Fig. 8.15