# pluriharmonic function

###### Definition.

Let $f:G\subset {\u2102}^{n}\to \u2102$ be a ${C}^{2}$ (twice continuously differentiable) function. $f$ is called pluriharmonic if for every complex line $\{a+bz\mid z\in \u2102\}$ the function $z\mapsto f(a+bz)$ is harmonic on the set $\{z\in \u2102\mid a+bz\in G\}$.

Note that every pluriharmonic function is a harmonic function, but not the other way around. Further it can be shown that for holomorphic functions^{} of several complex variables the real (and the imaginary) parts are locally pluriharmonic functions. Do note however that just because a function is harmonic in each variable separately does not imply that it is pluriharmonic.

## References

- 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title | pluriharmonic function |
---|---|

Canonical name | PluriharmonicFunction |

Date of creation | 2013-03-22 14:29:01 |

Last modified on | 2013-03-22 14:29:01 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 7 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 31C05 |

Classification | msc 32A50 |

Classification | msc 31C10 |

Synonym | pluriharmonic |