# identity theorem of power series

If the radii of convergence (http://planetmath.org/RadiusOfConvergence) of the power series^{} ${\sum}_{n=0}^{\mathrm{\infty}}{a}_{n}{(z-{z}_{0})}^{n}$ and ${\sum}_{n=0}^{\mathrm{\infty}}{b}_{n}{(z-{z}_{0})}^{n}$ are positive and the sums of the series are equal in infinitely many points which have ${z}_{0}$ as an accumulation point^{}, then the both series are identical, i.e. ${a}_{n}={b}_{n}$ for each $n=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots}$

Title | identity theorem of power series |
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Canonical name | IdentityTheoremOfPowerSeries |

Date of creation | 2013-03-22 16:47:08 |

Last modified on | 2013-03-22 16:47:08 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 4 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 30B10 |

Classification | msc 40A30 |

Related topic | IdentityTheoremOfHolomorphicFunctions |

Related topic | TheoremsOnComplexFunctionSeries |