Definition:Radius of Convergence/Complex Domain

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

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

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