Definition:Radius of Convergence/Complex Domain
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This page is about Radius of Convergence in the context of Power SeriesRadius. For other uses, see Radius of Convergence.
Definition
Let $\xi \in \C$ be a complex number.
For $z \in \C$, let:
- $\ds \map f z = \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n$
be a power series about $\xi$.
The radius of convergence is the extended real number $R \in \overline \R$ defined by:
- $R = \ds \inf \set {\cmod {z - \xi}: z \in \C, \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n \text{ is divergent} }$
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 $\map f z$ is absolutely convergent
- If $\cmod {z - \xi} > R$, then the power series defining $\map f z$ 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
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): radius of convergence