"determine the form of rational function in a plane hat has a positive value on unit circle."

Akash Matthews

Akash Matthews

Answered question

2022-02-16

"determine the form of rational function in a plane hat has a positive value on unit circle."
hint suggested me that such a rational function must have same number of zeros and poles inside unit disk. But I cannot understand that. Using symmetry(Schwarz reflection principle) I can see that number of poles(zeros) inside unit disk is equal to number of poles(zeros) outside unit disk. BUT I CANNOT SEE WHY NUMBER OF POLES AND ZEROS MUST BE EQUAL..
Who knows?

Answer & Explanation

Alexandra Haynes

Alexandra Haynes

Beginner2022-02-17Added 10 answers

Let g(t)=log(f(eit)) for tϵ[0,2π]. Then g is well-defined, real valued, g(0)=g(2π) and
g(t)=if(eit)eitf(eit).
Let γ denote the unit circle then
0=12πi02πg(t)dt=12πiγf(z)f(z)dz
and the latter is equal to the number of zeroes minus the number of poles of f on the unit disc.

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