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

Series of complex functions

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

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

which depends on ${\displaystyle z}$ 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

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

is called convergent in ${\displaystyle z_{0}}$ iff

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

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

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

exists.

Within the interior of the domain of convergence, the complex differentiability of all the ${\displaystyle f_{n}}$ 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 ${\displaystyle f_{n}(x)=c_{n}x^{n}}$, where ${\displaystyle c_{0},c_{1},c_{2},\ldots }$ is a sequence of constants.