# Hurwitz’s theorem

Define the ball at ${z}_{0}$ of radius $r$ as $$ and $D({z}_{0},r)=\{z\in G:|z-{z}_{0}|\le r\}$ is the closed ball^{} at ${z}_{0}$ of radius $r$.

###### Theorem (Hurwitz).

Let $G\mathrm{\subset}\mathrm{C}$ be a region and suppose the sequence
of holomorphic functions^{} $\mathrm{\{}{f}_{n}\mathrm{\}}$ converges uniformly on compact subsets
of $G$ to a holomorphic function $f$. If $f$ is not identically zero,
$D\mathit{}\mathrm{(}{z}_{\mathrm{0}}\mathrm{,}r\mathrm{)}\mathrm{\subset}G$ and $f\mathit{}\mathrm{(}z\mathrm{)}\mathrm{\ne}\mathrm{0}$ for $z$ such that
$\mathrm{|}z\mathrm{-}{z}_{\mathrm{0}}\mathrm{|}\mathrm{=}r$, then there exists an $N$ such that for all $n\mathrm{\ge}N$
$f$ and ${f}_{n}$ have the same number of zeros in $B\mathit{}\mathrm{(}{z}_{\mathrm{0}}\mathrm{,}r\mathrm{)}$.

What this theorem says is that if you have a sequence of holomorphic functions which converge uniformly on compact subsets (such a sequence always converges^{} to a holomorphic function but that’s another theorem altogether), the
function is not identically zero and furthermore the function is not
zero on the boundary of some ball,
then eventually
the functions of the sequence have the same number of zeros inside this ball as does the function.

Do note the requirement for $f$ not being identically zero. For example the sequence ${f}_{n}(z):=\frac{1}{n}$ converges uniformly on compact subsets to $f(z):=0$, but ${f}_{n}$ have no zeros anywhere, while $f$ is identically zero.

Also in general this result holds for bounded^{} convex subsets (http://planetmath.org/ConvexSet) but it is most
useful to for balls.

An immediate consequence of this theorem is this useful corollary.

###### Corollary.

If $G$ is a region and a sequence of holomorphic functions $\mathrm{\{}{f}_{n}\mathrm{\}}$ converges uniformly on compact subsets of $G$ to a holomorphic function $f$, and furthermore if ${f}_{n}$ never vanishes (is not zero for any point in $G$), then $f$ is either identically zero or also never vanishes.

## References

- 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.

Title | Hurwitz’s theorem |
---|---|

Canonical name | HurwitzsTheorem |

Date of creation | 2013-03-22 14:17:55 |

Last modified on | 2013-03-22 14:17:55 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 5 |

Author | jirka (4157) |

Entry type | Theorem |

Classification | msc 30C15 |

Related topic | CompositionAlgebraOverAlgebaicallyClosedFields |

Related topic | CompositionAlgebrasOverMathbbR |

Related topic | CompositionAlgebrasOverFiniteFields |

Related topic | CompositionAlgebrasOverMathbbQ |