Complex numbers - hurwitz theorem

In summary, the conversation is about solving a question regarding an open connected set and a sequence of functions converging uniformly. The conversation also includes a hint to use Hurwitz's theorem to find a suitable sequence and the participants expressing their lack of understanding and asking for clarification.
  • #1
hermanni
25
0
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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  • #2
Do you understand what Hurwitz's theorem tells you in this context?
 
  • #3
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.
 

Related to Complex numbers - hurwitz theorem

1. What is the Hurwitz theorem?

The Hurwitz theorem is a mathematical theorem in complex analysis that provides a necessary and sufficient condition for a sequence of complex numbers to converge to a limit in the complex plane. It was formulated by Adolf Hurwitz in 1915.

2. What is the significance of the Hurwitz theorem?

The Hurwitz theorem is important in complex analysis as it allows for the study of the convergence of sequences in the complex plane. This is useful in many applications, including physics, engineering, and computer science.

3. Can the Hurwitz theorem be applied to infinite series?

Yes, the Hurwitz theorem can be applied to infinite series. It provides a necessary and sufficient condition for an infinite series of complex numbers to converge to a limit in the complex plane.

4. How is the Hurwitz theorem related to the Cauchy convergence criterion?

The Hurwitz theorem is closely related to the Cauchy convergence criterion. In fact, the Hurwitz theorem is a generalization of the Cauchy convergence criterion for complex numbers.

5. Are there any limitations to the Hurwitz theorem?

One limitation of the Hurwitz theorem is that it only applies to sequences of complex numbers that converge in the complex plane. It does not provide information about sequences that diverge or converge to a limit outside of the complex plane.

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