Radius of Convergence - Radius of Convergence in Complex Analysis

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.

Read more about this topic:  Radius Of Convergence

Famous quotes containing the words complex and/or analysis:

    Uneducated people are unfortunate in that they do grasp complex issues, educated people, on the other hand, often do not understand simplicity, which is a far greater misfortune.
    Franz Grillparzer (1791–1872)

    The spider-mind acquires a faculty of memory, and, with it, a singular skill of analysis and synthesis, taking apart and putting together in different relations the meshes of its trap. Man had in the beginning no power of analysis or synthesis approaching that of the spider, or even of the honey-bee; but he had acute sensibility to the higher forces.
    Henry Brooks Adams (1838–1918)