Theorem 4.17.  Suppose the function    has radius of convergence  .  Then

(i)      is infinitely differentiable for all .  In fact

(ii)   for all  k,   ;   and

(iii)    where    denotes the derivative of f.  (When ,   denotes the function itself so that for all z.)

Proof.

Proof of Theorem 4.17 is in the book.

Complex Analysis for Mathematics and Engineering