Complex Power Series

Definition: Complex Power Series

A complex power series is an expression of the form

$$\sum _{n\in \mathcal{D}}{a}_n(x-c{)}^n,$$

where $(a_n{)}_{n\in \mathcal{D}}$ is a complex sequence and $c\in \mathbb{C}$.

INTUITION

By plugging in a concrete value $x^{\ast }\in \mathbb{C}$ for $x$, one obtains the complex series

$$\sum _{n\in \mathcal{D}}{a}_n(x^{\ast }-c{)}^n$$