1550251515-Classical_Complex_Analysis__Gonzalez_

Singularities/Residues/ Applications

Hence, by Exercises 4.2(26), we get the special limit

2. Consider the function

nn

n-+oo lira --n!en =^0

w = z^3 + 3z

755

(9.16-9)

for which w(O) = 0 and w'(O) = 3. We wish to express z as a function of
win some neighborhood of w = 0. Formula (9.16-7) gives

bn = 1 J d(

27l"in (n( (2 + 3)n

c+

Clearly, we can take for c+ any circle ( = re it, 0 :::; t :::; 271" with r < VS.

Since

the residue of the integrand at ( = 0 is zero if n is even. However, if
n = 2m + 1, the residue is given by the coefficient of (-^1 , i.e., for the value

of k satisfying 2k - n = -1, namely, k =^1 / 2 (n - 1) = m. Hence

and b2m+1 = (2m +^1 1)33m+l (-2m m -1)

(-1r (3m)!

·= 33 m+l (2m + l)!m!

Thus we obtain

(9.16-10)

where F denotes the hypergeometric function of Section 8.22. The series

in (9.16-10) converges absolutely on lwl < 2.

Theorem 9.25 can be generalized as follows. For simplicity we assume
that zo = wo = 0.

Theorem 9.26 (General Inversion Theorem). Suppose that the function
w = f(z) is analytic at the origin and has at that point a zero of order

m > 1. Then there are neighborhoods N 0 (0) and Ne(O), in the z-and

w-planes, respectively, such that f(z) is analytic on N 0 (0), and such that
to every point w ':f 0 in Ne(O) there corresponds exactly m different points