The identity theroem complex analysis

In summary, the conversation discusses the proof that there is no holomorphic function f in the open unit disk such that f(1/n)=((-1)^n)/(n^2) for n=2,3,4... The identity theorem is mentioned as a potential approach to the proof, and the function g(z)=(-1)^{\frac{1}{z}}z^2 is suggested as a possible solution.
  • #1
Scousergirl
16
0

Homework Statement


Prove that there is no holomorphic function f in the open unit disk such that f(1/n)=((-1)^n)/(n^2) for n=2,3,4...


Homework Equations


The identity theorem: Let f and g be holomorphic functions in the connected open subset of C, G. If f(z)=g(z) for all z in a subset of G that has a limit point in G, then f=g.



The Attempt at a Solution



We proved in an earlier example that there is not holomorphic function f in the open unit disk such that f(1/n)=2^(-n) n=2,3,4... Is the above function identical to this? I thought this was the route to go but i am not sure anymore. I know we have to somehow use the identity theorem.
 
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  • #2
Maybe it helps to consider the function [itex]g(z)=(-1)^{\frac{1}{z}}z^2[/itex]
 

FAQ: The identity theroem complex analysis

What is the identity theorem in complex analysis?

The identity theorem in complex analysis states that if two complex functions are equal on a non-empty open subset of the complex plane, then they are equal everywhere on the complex plane. In simpler terms, if two complex functions have the same values on a certain region of the complex plane, then they must be the same function.

Why is the identity theorem important in complex analysis?

The identity theorem is important because it allows us to prove the uniqueness of a complex function. By showing that two functions are equal on a certain region, we can conclude that they are equal everywhere. This is a powerful tool in complex analysis and is used in various proofs and theorems.

How is the identity theorem used in practice?

In practice, the identity theorem is used to prove the uniqueness of solutions to complex equations. It is also used to show that two complex functions are equal by finding a region where they have the same values. Additionally, the identity theorem is used to prove other important theorems in complex analysis.

Are there any limitations to the identity theorem?

One limitation of the identity theorem is that it only applies to analytic functions, which are functions that can be expressed as power series. It also only applies to open subsets of the complex plane, so it cannot be used on functions with singularities or on the boundary of a region.

Can the identity theorem be extended to higher dimensions?

Yes, the identity theorem can be extended to higher dimensions in complex analysis. In fact, the theorem holds in any number of dimensions, as long as the functions are analytic and the region is open. This extension is known as the identity principle and is an important concept in higher dimensional complex analysis.

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