Analyticity of Complex Function f(z)=1/(z^2-1)

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Discussion Overview

The discussion centers on the analyticity of the complex function f(z) = 1/(z^2 - 1). Participants explore whether the function is analytic and under what conditions, focusing on its poles and the implications of complex differentiability.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants assert that the function is analytic on the set {z | z ≠ 1 and z ≠ -1}, indicating it is analytic away from its poles.
  • Others request clarification on how the function is shown to be analytic on that set, seeking a demonstration of the underlying principles.
  • One participant explains that complex differentiability implies analyticity, noting that the function's power series converges on the largest open disc that does not include its singularities at z = 1 and z = -1.
  • Another participant suggests that the algebra of differentiation remains consistent between real and complex analysis, implying that the methods used in real analysis can be applied here.
  • There is mention of a method to expand the function around points other than the poles using series expansion techniques.

Areas of Agreement / Disagreement

Participants generally agree that the function is analytic away from its poles, but there is no consensus on the specific methods or explanations for demonstrating this analyticity.

Contextual Notes

Some assumptions about the nature of complex differentiability and the conditions under which the function is analytic are not fully explored, leaving some mathematical steps unresolved.

tthivanka
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f(z)=1/(z^2-1)
Is this function analytical or not
 
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It is analytic on the set {z | z =/= 1 and z =/= -1}, that is, it is analytic away from its poles.
 
could u please show me how It is analytic on the set {z | z =/= 1 and z =/= -1}
 
Sure, the simplest way is to realize that nothing has really changed in this problem from real analysis. Although complex differentiability is much more powerful than real differentiability, the algebra of taking derivatives is still the same. Thus we can differentiate polynomials and rational functions just as we did in calculus, and this would immediately give you the answer.
 
tthivanka said:
could u please show me how It is analytic on the set {z | z =/= 1 and z =/= -1}

A function that is complex differentiable on an open set is analytic on that set and its power series converges on the largest open disc that does not contain a singularity/pole. The singuarities of this function are at z = 1 and z = -1.
 
They are using a big theorem that differentiable implies analytic. It is also elementary to expand this function about any point a other than 1,-1, by setting z = (z-a+a), expanding z^2 in terms of (z-a), and then using the geometric series. I.e. when c is not zero, 1/[c + f(z-a)] equals (1/c)[1/{1 - (-1/c)(f(z-a))}], and you know how to expand 1/(1-anything) as 1 + anything + (anything)^2+...
 

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