# harmonic function

A twice-differentiable real or complex-valued function $f:U\to \mathbb{R}$ or $f:U\to \u2102$, where $U\subseteq {\mathbb{R}}^{n}$ is some , is called *harmonic* if its Laplacian vanishes on $U$, i.e. if

$$\mathrm{\Delta}f\equiv 0.$$ |

Any harmonic function^{} $f:{\mathbb{R}}^{n}\to \mathbb{R}$ or $f:{\mathbb{R}}^{n}\to \u2102$ satisfies Liouville’s theorem. Indeed, a holomorphic function^{} *is* harmonic, and a real harmonic function $f:U\to \mathbb{R}$, where $U\subseteq {\mathbb{R}}^{2}$, is locally the real part of a holomorphic function. In fact, it is enough that a harmonic function $f$ be below (or above) to conclude that it is .

Title | harmonic function |

Canonical name | HarmonicFunction |

Date of creation | 2013-03-22 12:43:46 |

Last modified on | 2013-03-22 12:43:46 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 9 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 31C05 |

Classification | msc 31B05 |

Classification | msc 31A05 |

Classification | msc 30F15 |

Related topic | RadosTheorem |

Related topic | SubharmonicAndSuperharmonicFunctions |

Related topic | DirichletProblem |

Related topic | NeumannProblem |