1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 537

8.4 Operations with Power Series

In the preceding section we have seen that there are cases in which the
power series expansion of an analytic function is more easily obtained from
those of some other related functions by subjecting such functions and
their expansions to certain algebraic or analytic manipulations. This is
particularly true in the case of those functions for which there is no simple
way of obtaining a general formula for their nth derivatives, as required
by the Cauchy-Tay.lor theorem.
To lay the groundwork for some further important examples of the
method, we proceed to discuss some theorems concerning operations with
power series. For simplicity we consider only series in powers of z since
series in powers of z - a can be reduced to the former by the translation
z - a = z'.

Theorem 8.6 Suppose that J(z) = L::'=o anzn and g(z) = L::'=o bnzn
have radii of convergence Ri and Rz, respectively, and let ll' and /3 be any
complex constants. Then
00
af(z) + f3g(z) = :L)aan + f3bn)zn (8.4-1)
n=O

where the power series on the right converges at least for lzl < R =
min(R1, Rz).

Proof For any z such that izl < R, let

Then

N

SN(z) = L anzn

n=O

and

N

N

TN(z) = L bnzn

n=O

aSN(z) + /3TN(z) = L(aan + f3bn)zn

n=O

and if follows that

00

lim [aSN(z) + f3TN(z)] = af(z) + f3g(z) = L(aan + f3bn)zn

N-+oo n=O

In particular, we have

00

J(z) ± g(z) = L(an ± bn)zn

n=O

for izl < R.