1550251515-Classical_Complex_Analysis__Gonzalez_

538 Chapter 8

Note 'rhe radius of convergence of 2:::"= 0 (aan + f3bn)zn can be.greater
than R = min(Ri, R 2 ). For instance, we have

yet

1 00

l-z = Lzn

n=O

for lzl < 1

for lzl < 1

·1 z

00

1

--- = L 2nzn

1 - Z (2 - Z )(1 - Z) n=O

for lzl < 2

the function on the left having a removable singularity at z = 1.

Theorem 8.7 Suppose that J(z) = E::"=o anzn for lzl < Ri and g(z) =
E::"=o bnzn for lzl < R2·· Then
00
f(z)g(z) = L CnZn (8.4-2)
n=O
where Cn = aobn + aibn-1 + · · · + anbo = E~=O akbn-k, the power series
on the right of (8.4-2) converging at least for lzl < R = min(R 1 , R 2 ).

Proof For any particular value of z such that lzl < R both series

E::"=o anzn and L:::"=o bnzn converge absolutely. Hence (8.4-2) follows from

Corollary 4.3.

Note Again, in this case the radius of conve~gence of E::"==o cnzn can be

greater than R = min(R 1 ,R 2 ). For example, we have

I+z 2 n

:--= 1+2z + 2z + · · · + 2z + · · · for Jzl < 1

1-z

and

for lzJ < 1

The coefficients of the product of the two series are c 0 = 1, c1 = c 2 = · · · =

en = 0 for all n 2:: 1, so that E::"=o CnZn = 1, which converges for all z.

Of course, the product of the two given functions is also 1, provided that

the singularities z = 1 and z = -1 are removed according to Riemann's

theorem.

Theorem 8.8 Suppose that J(z) = E::"=o anzn for Jzl < R 1 and g(z) =
E::"=o bnzn for Jzl < R2 with bo -:/= O. Let p = d(O, K) where K is the set of