Complex Power Series

Definition: Let $(a_n)_{n=0}^{\infty}$ be a sequence of complex numbers. The corresponding Complex Power Series centered at $z_0 \in \mathbb{C}$ is denoted $\displaystyle{S^{z_0}(z) = \sum_{n=0}^{\infty} a_n(z - z_0)^n}$.

It is important to note that $S^{z_0}(z)$ always converges at $z = z_0$.

We can immediately reformulate the extremely important Weierstrass M-test mentioned on the Sequences and Series of Complex Functions page to power series.

Theorem 1 (The Weierstrass M-Test for Complex Power Series): Consider the power series $\displaystyle{\sum_{n=0}^{\infty} a_n(z - z_0)^n}$. If there exists a sequence of nonnegative real numbers $(M_n)_{n=0}^{\infty}$ such that $\mid a_n(z - z_0)^n \mid \leq M_n$ for all $z \in A$ and for each $n \in \{ 0, 1, 2, ... \}$ and such that $\displaystyle{\sum_{n=0}^{\infty} M_n}$ converges then the power series $\displaystyle{\sum_{n=0}^{\infty} a_n(z_0)^n}$ converges uniformly on $A$.

For example, consider the following complex power series:

(1)

\begin{align} \quad \sum_{n=0}^{\infty} z^n \end{align}

Let $A = D(0, 1)$. Then $0 \leq \mid z^n \mid = r < 1$ for all $z \in A$. But we know that if $0 < r < 1$ then the series $\displaystyle{\sum_{n=0}^{\infty} r^n}$ converges as a geometric series, and so $\displaystyle{\sum_{n=0}^{\infty} z^n}$ converges whenever $\mid z \mid < 1$. This series is significant and is often referred to as the complex geometric series.

Of course, we can form many other complex power series. For example:

(2)

\begin{align} \quad \sum_{n=0}^{\infty} \frac{(z - z_0)^n}{n!} \end{align}

This is a perfectly valid complex power series though we will need to develop a richer theory to determine whether this complex power series converges or not.