1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 525

Proof Letting Fn(z) = f1(z)+· · + fn(z), the functions Fn(z) (n = 1, 2, ... )

satisfy the conditions of Theorem 8.2. Hence

F(z) = n->oo lim Fn(z)

is analytic in A, and

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the convergence being uniform on compact subsets of A.

This corollary allows the termwise differentiation of a series of analytic

functions provided that the series converges in some open set, and uniformly

on compact subsets.

Example Let F(z) = E::"=o anzn for lzl < r. We have seen that the series

converges uniformly on lzl :::; r1 < r, and so on every compact subset of

the open disk lzl < r. Then it follows that

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F'(z) = L nanzn-I for lzl < r

n=l

F"(z) = L n(n - l)anzn-^2 for lzl < r

n=2

etc.

Remark The property stated in Corollary 8.5 is in general false for series
of real analytic functions.

Example The series I:::°= 1 (sin n^2 x) / n^2 converges uniformly on the real
line, yet the derived series E::"=l cosn^2 x does not converge for any x (since
the general term does not tend to zero as n ~ oo ).

Theorem 8.3 Suppose that:

1. The functions f n ( z) ( n = 1, 2, ... ) are analytic in some open set A.

2. The series E::"=l lfn(z)I converges uniformly on every compact subset
   of A.

Then the series E::"=l IJAk)(z)I converges uniformly on every compact
subset of A.
We note that this theorem cannot be derived from Corollary 8.5, since
the terms in E::"=i lfn(z)I are real functions of z.