1550251515-Classical_Complex_Analysis__Gonzalez_

754 Chapter^9

we assumed that z 0 = w 0 = 0). There we saw that the coefficients b 1 , b 2 ,

... can be obtained successively in terms of rational operations involving

the coefficients a 1 , a 2 , •••• However, in (9.16-4) we gave a general formula
for bn. This formula can be put in the more convenient form

(9.16-7)

after integration by parts.

Examples 1. Consider the function

w = f(z) = ze-z

We wish to express z = f-^1 (w) in terms of a series in powers of w in

some neighborhood of the origin. We note that in this example f(O) = 0

and f'(O) = 1. Formula (9.16-7) gives

1 J en( d( 1 dn-l nn-l

nbn = 27ri ~ = (n -1)! dzn-1 enz lz=o = (n -1)!

c+

Hence bn = nn-l /n! and we obtain

(^00) n-1
z=l:~wn
n=l n.
(9.16-8)
By using the ratio test it is easy to see that (9.16-8) converges absolutely

for lwl < l/e ~ 0.36

Note In (9.16-8) suppose that w = u is real and such that lul < l/e.

Then z is real, say z = x, and the equation u = xe-x shows that u --t e-^1

as x --t 1, and conversely. Since the series

(^00) n-1
L
--e n -n
n!
n=l
converges by the Raabe-Duhamel test, Abel's lirrii.t theorem yields
(^00) n-1
l = L :!en
n=l
This series has positive 'terms and