Complex Analysis/Function series, power series, Euler's formula, polar form, argument

Series of complex functions Edit

Given a sequence of (possibly holomorphic functions)   we can, in the case of convergence, form the series


which depends on   and may be seen as a complex function, where domain of definition equals domain of convergence. We first note the obvious definitions (we count from zero since this will allow for the important special case of power series without any modifications).

Definition 3.1:

The series


is called convergent in   iff


exists. It is called absolutely convergent in   iff



Within the interior of the domain of convergence, the complex differentiability of all the   implies the differentiability of the corresponding series, and moreover we may differentiate term-wise.

Theorem 3.2:

Power series Edit

In this section, we specialize the considerations of the previous section down to the case  , where   is a sequence of constants.