# p-adic analytic

###### Definition.

Let ${\mathrm{C}}_{p}$ be the field of complex $p$-adic numbers (http://planetmath.org/ComplexPAdicNumbers). Let $U$ be a domain in ${\mathrm{C}}_{p}$. A function^{} $f\mathrm{:}U\mathrm{\u27f6}{\mathrm{C}}_{p}$ is $p$-adic analytic^{} if $f$ has a Taylor series^{} (with coefficients in ${\mathrm{C}}_{p}$) about each point $z\mathrm{\in}U$ that converges to the function $f$ in an open neighborhood of $z$.

For example, the $p$-adic exponential function^{} (http://planetmath.org/PAdicExponentialAndPAdicLogarithm) is analytic on its domain of definition:

$$ |

The study of $p$-adic analytic functions is usually called $p$-adic analysis^{} and it is very similar to complex analysis in many respects, although there are important differences^{} coming from the distinct topologies of $\u2102$ and ${\u2102}_{p}$.

Title | p-adic analytic |
---|---|

Canonical name | PadicAnalytic |

Date of creation | 2013-03-22 15:13:53 |

Last modified on | 2013-03-22 15:13:53 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 4 |

Author | alozano (2414) |

Entry type | Definition |

Classification | msc 11S99 |

Classification | msc 12J12 |

Classification | msc 11S80 |

Synonym | $p$-adic analytic |

Related topic | Analytic |

Related topic | PAdicExponentialAndPAdicLogarithm |

Defines | $p$-adic analysis |

Defines | p-adic analysis |