# Definition:Radius of Convergence/Complex Domain

< Definition:Radius of Convergence(Redirected from Definition:Radius of Convergence of Complex Power Series)

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*This page is about the radius of convergence of a power series. For other uses, see Definition:Radius.*

## Contents

## Definition

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

For $z \in \C$, let:

- $\displaystyle f \paren z = \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n$

be a power series about $\xi$.

The **radius of convergence** is the extended real number $R \in \overline \R$ defined by:

- $R = \displaystyle \inf \set {\cmod {z - \xi}: z \in \C, \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n \text{ is divergent} }$

where a divergent series is a series that is not convergent.

As usual, $\inf \O = +\infty$.

## Also see

- Existence of Radius of Convergence of Complex Power Series, which shows that:

- If $\cmod {z - \xi} < R$, then the power series defining $f \paren z$ is absolutely convergent
- If $\cmod {z - \xi} > R$, then the power series defining $f \paren z$ is divergent.

## Linguistic Note

The plural of **radius** is **radii**, pronounced ** ray-dee-eye**.

This irregular plural form stems from the Latin origin of the word **radius**, meaning **ray**.

The ugly incorrect form **radiuses** can apparently be found, but rarely in a mathematical context.

## Sources

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