Radius of Convergence in Complex Analysis
A power series with a positive radius of convergence can be made into a holomorphic function by taking its argument to be a complex variable. The radius of convergence can be characterized by the following theorem:
- The radius of convergence of a power series f centered on a point a is equal to the distance from a to the nearest point where f cannot be defined in a way that makes it holomorphic.
The set of all points whose distance to a is strictly less than the radius of convergence is called the disk of convergence.
The nearest point means the nearest point in the complex plane, not necessarily on the real line, even if the center and all coefficients are real. For example, the function
has no singularities on the real line, since has no real roots. Its Taylor series about 0 is given by
The root test shows that its radius of convergence is 1. In accordance with this, the function ƒ(z) has singularities at ±i, which are at a distance 1 from 0.
For a proof of this theorem, see analyticity of holomorphic functions.
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