Rational Functions: Analysis & Agreement

[Pere Callahan]

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

 

[Pere Callahan]

Anyone?

 

[Pere Callahan]

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.

 

[AndrewDSmith]

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.