# holomorphic

Let $U\subset \u2102$ be a domain in the complex numbers^{}. A
function $f:U\u27f6\u2102$ is *holomorphic* if $f$
has a complex derivative^{} at every point $x$ in $U$, i.e. if

$$\underset{z\to {z}_{0}}{lim}\frac{f(z)-f({z}_{0})}{z-{z}_{0}}$$ |

exists for all ${z}_{0}\in U$.

More generally, if $\mathrm{\Omega}\subset {\u2102}^{n}$ is a domain, then a function $f:\mathrm{\Omega}\to \u2102$ is said to be *holomorphic* if $f$ is holomorphic in each of the variables. The class of all holomorphic functions on $\mathrm{\Omega}$ is usually denoted by $\mathcal{O}(\mathrm{\Omega})$.

Title | holomorphic |

Canonical name | Holomorphic |

Date of creation | 2013-03-22 12:04:33 |

Last modified on | 2013-03-22 12:04:33 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 12 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 30D20 |

Classification | msc 32A10 |

Synonym | holomorphic function |

Synonym | regular function |

Synonym | complex differentiable |

Related topic | CauchyRiemannEquations |

Related topic | Analytic |