Hurwitz theorem (function theory)

The Hurwitz theorem (named after Adolf Hurwitz (1859–1919)) is a theorem from function theory .

Hurwitz's theorem

Let be a sequence of functions that are holomorphic in a domain and the sequence converges compactly to . If the number of zeros of the functions (from an index) is limited by, then the following applies: ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle G}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f}$ ${\displaystyle f_{n}}$ ${\displaystyle M}$

The limit function has a maximum of zeros on or on ( is the zero function ). ${\displaystyle f}$ ${\displaystyle M}$ ${\displaystyle G}$ ${\displaystyle f\equiv 0}$ ${\displaystyle G}$ ${\displaystyle f}$

Inference

Let be a sequence of functions that are holomorphic and injective in a domain , and the sequence converges compactly to . ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle G}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f}$

Then is injectively on or constant on . ${\displaystyle f}$ ${\displaystyle G}$ ${\displaystyle G}$

Web links

• Eric W. Weisstein : Hurwitz's theorem . In: MathWorld (English).
• Jiri Lebl: Hurwitz's theorem . In: PlanetMath . (English)