Definition:Power Series/Complex Domain

Definition

Let $\xi \in \C$ be a complex number.

Let $\sequence {a_n}$ be a sequence in $\C$.

The series $\ds \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n$, where $z \in \C$ is a variable, is called a (complex) power series in $z$ about the point $\xi$.

Examples

Example: $\ds \sum_{n \mathop \ge 0} n z^n$

The complex power series:

$S = \ds \sum_{n \mathop \ge 0} n z^n$

has a radius of convergence of $1$.

Example: $\ds \sum_{n \mathop \ge 0} \dfrac {3^n - 1} {2^n + 1} z^n$

Let $\sequence {a_n}$ be the sequence defined as:

$a_n = \dfrac {3^n - 1} {2^n + 1}$

The complex power series:

$S = \ds \sum_{n \mathop \ge 0} a_n z^n$

has a radius of convergence of $\dfrac 2 3$.

Example: $\ds \sum_{n \mathop \ge 0} \dfrac {\paren {2 n}!} {\paren {n!}^2} z^n$

Let $\sequence {a_n}$ be the sequence defined as:

$a_n = \dfrac {\paren {2 n}!} {\paren {n!}^2} z^n$

The complex power series:

$S = \ds \sum_{n \mathop \ge 0} a_n z^n$

has a radius of convergence of $\dfrac 1 4$.

Example: $\ds \sum_{n \mathop \ge 0} \dfrac {\cos i n} {n^2} z^n$

Let $\sequence {a_n}$ be the sequence defined as:

$a_n = \dfrac {\cos i n} {n^2} z^n$

The complex power series:

$S = \ds \sum_{n \mathop \ge 0} a_n z^n$

has a radius of convergence of $\dfrac 1 e$.

Also see

• Results about complex power series can be found here.

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.4$. Power Series: $(4.10)$