# radius of convergence of a complex function

Let $f$ be an analytic function^{} defined in a disk of radius $R$ about a point ${z}_{0}\in \u2102$. Then the radius of convergence^{} of the Taylor series^{} of $f$ about ${z}_{0}$ is at least $R$.

For example, the function $a(z)=1/{(1-z)}^{2}$ is analytic inside the disk $$. Hence its the radius of covergence of its Taylor series about $0$ is at least $1$. By direct examination of the Taylor series we can see that its radius of convergence is, in fact, equal to $1$.

Colloquially, this theorem is stated in the sometimes imprecise but memorable form “The radius of convergence of the Taylor series is the distance to the nearest singularity.”

Title | radius of convergence of a complex function |
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Canonical name | RadiusOfConvergenceOfAComplexFunction |

Date of creation | 2013-03-22 14:40:33 |

Last modified on | 2013-03-22 14:40:33 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 6 |

Author | rspuzio (6075) |

Entry type | Theorem |

Classification | msc 30B10 |