Rational Functions: Analysis & Agreement

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Discussion Overview

The discussion revolves around the properties of rational functions, specifically whether two rational functions that coincide on the unit circle also coincide everywhere in the complex plane. Participants explore the implications of this condition and consider a related question about the functions having the same modulus on the unit circle.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant argues that if two rational functions coincide on the unit circle and have no poles on it, they must coincide everywhere in the complex plane due to the uniqueness principle for analytic functions.
  • The same participant notes that the poles of both functions must be the same for this conclusion to hold, as differing poles would lead to different limits at those points.
  • Another participant questions the conclusion when only the modulus of the functions is the same on the unit circle, expressing uncertainty about how to apply the previous reasoning since modulus is not an analytic function.
  • A different participant asserts that the original claim about coincidence is true and suggests using the values of the difference of the functions on a convergent sequence to show that they must be equal everywhere.
  • This participant also states that the claim about the same modulus leading to coincidence is false, providing the example of the functions z and -z to illustrate this point.

Areas of Agreement / Disagreement

There is disagreement regarding the implications of having the same modulus on the unit circle, with one participant asserting that it does not guarantee coincidence everywhere, while another participant maintains that the original claim about coincidence based on agreement on the unit circle is valid.

Contextual Notes

Participants have not resolved the implications of the modulus condition, and there are assumptions regarding the nature of the poles that have not been fully explored.

Pere Callahan
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Hi,

I was wondering whether two rational functions f,g whch coincide on the unit circle actually coincide on all of C.


I would say yes. Let D be the set of all complex numbers with the poles of both f and g removed (let's assume there are no poles on the unit circle). This is then open and connected, hence a domain and f and g are analytic there. Moreover they agree on the unit circle which is a set with at least one nonisolated point (in fact all points are nonisolated) and which lies in D, so the uniqueness principle implies that f and g agree on D.

But the poles have to be the same as well. For if w is a pole of f but not of g then the limit of f as z approaches w is infinity and must be the same as the limit of g as w approaches infinty, because a neighbourhood of z is contained in D.

Is this correct?

thanks
 
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Ok, I think my argument from above is correct.
But what if we only know that the two rational functions' modulus is the same on the unit circle?

Do they still have to coincide everywhere? I don't know how to adopt my previous reasining because the modulus is not an analytic function..

Thanks.
 
To your original question, the answer is yes. You can use the values of f-g on any convergent sequence to a point on the circle to expand in a series about that point - in that sequence the coefficients are all zero. QED

To your second question its not true - consider z and -z for example.
 

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