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

Series of complex functions

Given a sequence of (possibly holomorphic functions) $f_{0},f_{1},\ldots ,f_{n},f_{n+1},\ldots$ we can, in the case of convergence, form the series

$\sum _{n=0}^{\infty }f_{n}(z)$,

which depends on $z$ 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

$\sum _{n=0}^{\infty }f_{n}(z)$

is called convergent in $z_{0}$ iff

$\lim _{N\to \infty }\sum _{n=0}^{N}f_{n}(z_{0})$

exists. It is called absolutely convergent in $z_{0}$ iff

$\lim _{N\to \infty }\sum _{n=0}^{N}|f_{n}(z_{0})|$

exists.

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

Theorem 3.2:

Power series

In this section, we specialize the considerations of the previous section down to the case $f_{n}(x)=c_{n}x^{n}$, where $c_{0},c_{1},c_{2},\ldots$ is a sequence of constants.