# places of holomorphic function

If $c$ is a complex constant and $f$ a holomorphic function^{} in a domain $D$ of $\u2102$, then $f$ has in every compact^{} (closed (http://planetmath.org/TopologyOfTheComplexPlane) and bounded^{} (http://planetmath.org/Bounded)) subdomain of $D$ at most a finite set^{} of http://planetmath.org/node/9084$\mathrm{c}$-places, i.e. the points $z$ where $f(z)=c$, except when $f(z)\equiv c$ in the whole $D$.

*Proof.* Let $A$ be a subdomain of $D$. Suppose that there is an infinite^{} amount of $c$-places of $f$ in $A$. By http://planetmath.org/node/2125Bolzano–Weierstrass theorem, these $c$-places have an accumulation point^{} ${z}_{0}$, which belongs to the closed set^{} $A$. Define the constant function $g$ such that

$$g(z)=c$$ |

for all $z$ in $D$. Then $g$ is holomorphic in the domain $D$ and $g(z)=c$ in an infinite subset of $D$ with the accumulation point ${z}_{0}$. Thus in the $c$-places of $f$ we have

$$g(z)=f(z).$$ |

Consequently, the identity theorem of holomorphic functions implies that

$$f(z)=g(z)=c$$ |

in the whole $D$. Q.E.D.

Title | places of holomorphic function |
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Canonical name | PlacesOfHolomorphicFunction |

Date of creation | 2013-03-22 18:54:18 |

Last modified on | 2013-03-22 18:54:18 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 8 |

Author | pahio (2872) |

Entry type | Corollary |

Classification | msc 30A99 |

Related topic | ZerosAndPolesOfRationalFunction |

Related topic | IdentityTheorem |

Related topic | TopologyOfTheComplexPlane |