Definition:Power Series/Complex Domain
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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: $(4.10)$