Hurwitz's Theorem (complex Analysis)

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## Statement

## Remarks

## Applications

## Proof

## See also

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Hurwitz's Theorem Complex Analysis

In mathematics and in particular the field of complex analysis, **Hurwitz's theorem** is a theorem associating the zeroes of a sequence of holomorphic, compact locally uniformly convergent functions with that of their corresponding limit. The theorem is named after Adolf Hurwitz.

Let {*f _{k}*} be a sequence of holomorphic functions on a connected open set

The theorem does not guarantee that the result will hold for arbitrary disks. Indeed, if one chooses a disk such that *f* has zeroes on its boundary, the theorem fails. An explicit example is to consider the unit disk **D** and the sequence defined by

which converges uniformly to *f*(*z*) = *z* - 1. The function *f*(*z*) contains no zeroes in **D**; however, each *f _{n}* has exactly one zero in the disk corresponding to the real value 1 - (1/

Hurwitz's theorem is used in the proof of the Riemann mapping theorem,^{[2]} and also has the following two corollaries as an immediate consequence:

- Let
*G*be a connected, open set and {*f*} a sequence of holomorphic functions which converge uniformly on compact subsets of_{n}*G*to a holomorphic function*f*. If each*f*is nonzero everywhere in_{n}*G*, then*f*is either identically zero or also is nowhere zero. - If {
*f*} is a sequence of univalent functions on a connected open set_{n}*G*that converge uniformly on compact subsets of*G*to a holomorphic function*f*, then either*f*is univalent or constant.^{[2]}

Let *f* be an analytic function on an open subset of the complex plane with a zero of order *m* at *z*_{0}, and suppose that {*f _{n}*} is a sequence of functions converging uniformly on compact subsets to

Denoting the number of zeros of *f _{k}*(

In the above step, we were able to interchange the integral and the limit because of the uniform convergence of the integrand. We have shown that *N _{k}* ->

**^**Ahlfors 1966, p. 176, Ahlfors 1978, p. 178- ^
^{a}^{b}Gamelin, Theodore (2001).*Complex Analysis*. Springer. ISBN 978-0387950693. **^**Ahlfors 1966, p. 176, Ahlfors 1978, p. 178

- Ahlfors, Lars V. (1966),
*Complex analysis. An introduction to the theory of analytic functions of one complex variable*, International Series in Pure and Applied Mathematics (2nd ed.), McGraw-Hill - Ahlfors, Lars V. (1978),
*Complex analysis. An introduction to the theory of analytic functions of one complex variable*, International Series in Pure and Applied Mathematics (3rd ed.), McGraw-Hill, ISBN 0070006571 - John B. Conway.
*Functions of One Complex Variable I*. Springer-Verlag, New York, New York, 1978. - E. C. Titchmarsh,
*The Theory of Functions*, second edition (Oxford University Press, 1939; reprinted 1985), p. 119. - Solomentsev, E.D. (2001) [1994], "Hurwitz theorem",
*Encyclopedia of Mathematics*, EMS Press

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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