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Complex Analysis / Singularities / Argument Principle : Let f be meromorphic on the region G and not constant; show that neither the poles nor the zeros of f have a limit point in G.

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Let f be meromorphic on the region G and not constant; show that neither the poles nor the zeros of f have a limit point in G.

In your solution, please refer to theorems or certain lemmas. Justify your claims and steps. I want to learn not just have the right answer. Thanks.

https://brainmass.com/math/complex-analysis/complex-analysis-singularities-argument-principle-59166

Solution Preview

Hi, we can prove this indirect method.

Suppose f(z) has zero in G and z=a is one of them. Let C be an scro curve in D encircling z= a and not passing through any zero such that not other zero of f(z) lies either in Ci or on C.

Let E(epslon) be the minimum value of lf(z)l on C. Then by hypothesis E(epslon) >0. That is

l f(z) l >= E(epslon) for ...

Solution Summary

Meromorphic functions, poles and zeros are investigated. The solution is detailed and well presented.

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