Complex numbers - hurwitz theorem

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SUMMARY

The discussion centers on applying Hurwitz's theorem to a sequence of functions converging uniformly on compact subsets of an open connected set D. The user seeks clarification on how to utilize the theorem, particularly in the context of demonstrating that if f is nonconstant and has a zero at z, then the sequence of functions fn will also exhibit a zero at z for sufficiently large n. The hint provided suggests examining the behavior of f in a disk around z as the radius approaches zero, which is crucial for leveraging the theorem effectively.

PREREQUISITES
  • Understanding of complex analysis concepts, particularly uniform convergence.
  • Familiarity with Hurwitz's theorem and its implications for zeros of functions.
  • Knowledge of open connected sets in the context of complex functions.
  • Basic grasp of sequences and limits in mathematical analysis.
NEXT STEPS
  • Study the proof and applications of Hurwitz's theorem in complex analysis.
  • Explore uniform convergence and its significance in function analysis.
  • Investigate the properties of open connected sets in complex function theory.
  • Learn about the behavior of complex functions in neighborhoods of their zeros.
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Students and professionals in mathematics, particularly those specializing in complex analysis, as well as educators seeking to deepen their understanding of Hurwitz's theorem and its applications.

hermanni
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Hi all,
I'm trying to solve this question , can anyone help??
Suppose that D is an open connected set , fn ->f uniformly on compact subsets of D. If f is nonconstant and z in D , then there exists N and a sequence zn-> z such that
fn ( zn ) = f(z) for all n > N.

hint: assume that f(z) = 0. Apply Hurwitz theorem to in disk D(z0 , rj ) for a suitable sequence of rj -> 0

I reallt don't have an idea and I don't understand how to use hint.Can anyone give a hint??
 
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Do you understand what Hurwitz's theorem tells you in this context?
 
Actually I didn't . The theorem says fn and f have the same number of zeroes, I don't understand how we supposed to use it.
 

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