Rouché's Theorem in complex analysis on the relation of the number of zeros and poles of meromorphic functions in a region [closed]

Question

This question is from my son referenced in my earlier question, Need advice or assistance for son who is in prison. His interest is scattering theory . He asked me to post this question:

Hello and thanks to everyone for help finding papers thus far. I am currently looking for some further information and applications of Rouché's theorem in complex analysis on the relation of the number of zeros and poles of meromorphic functions in a region. I have the basic statement, but am looking for some more advanced or peripheral results, reformulations, extensions, etc. Any other theorems with conditions for the relation of the poles and zeros of two functions in a region would also be helpful. To be very specific, if $f=g+h$, with all functions meromorphic in the plane, I'm looking for conditions on $f$, $g$, $h$, so that $f$ and $g$ have the same number of poles and zeros in a region. The form of the particular functions I'm dealing with are generally highly oscillatory, nonlinear Fourier transforms of smooth, compactly supported functions where the nonlinearity can cause poles, but sometimes their real and imaginary parts can be controlled well, so conditions relating their arguments, or real/im parts might be useful. Thanks. -Travis.

Answer 1

Dear Mr. Cunningham:

The question you son asked is at an elementary level, but it has a wide range of associations. I think a one good reference is Prof. Tao's article:

https://terrytao.wordpress.com/2020/12/23/246b-notes-1-zeroes-poles-and-factorisation-of-meromorphic-functions/#more-12187

The article should be relatively self-contained. Hope it helps.

Some of the material are obviously covered in standard complex analysis texts (Consider https://www.amazon.com/Complex-Analysis-Princeton-Lectures-No/dp/0691113858/ref=sr_1_2?dchild=1&keywords=stein+complex+analysis&qid=1619136831&sr=8-2, for example).