Uniqueness Theorem: Complex Analysis Explained

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The discussion centers on the "Uniqueness Theorem" in complex analysis, specifically seeking clarification on its statement. The theorem asserts that if two analytic functions agree on a set with a limit point, they must be identical everywhere in their domain. An example is provided where an analytic function f, defined by f(1/n) = 1/n^2 for n = 1, 2, 3, is shown to equal g(z) = z^2 throughout the complex plane. The conversation emphasizes the need for specificity regarding what aspect of uniqueness is being referenced. Understanding this theorem is crucial for applying it to complex functions and their properties.
matheater
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Will anybody please tell me what is the statement of the "Uniqueness theorem" in Complex analysis??
 
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I'm not sure if there is a unique uniqueness theorem. Can you maybe be more specific? Uniqueness of what?
 
nicksauce said:
I'm not sure if there is a unique uniqueness theorem. Can you maybe be more specific? Uniqueness of what?
I have the sum:
Let f:C-->C be analytic s.t f(1/n)=1/n^2,n=1,2,3...
Then(out of 4 ans i am giving the correct one)f(z)=z^2 for all z in C
The solution is done as follows:
Let g(z)=z^2 for all z in C.Then f(z)=g(z) for all z in {1/n :n=1,2,3.}.since {1/n :n=1,2,3.} is an infinite set having a limit point "0",so by "uniqueness theorem" f(z)=g(z) for all z in C.

Here i want to have the statement of this Uniqueness theorem...
 

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