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Analytic function whose zeros are of finite order

Recall that a point z0 is an accumulation point of a set S if each deleted neighborhood of z0 contains at least one point in S.

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One form of the Bolzano-Weierstrass theorem can be stated as follows:

An infinite set of points lying in a closed bounded region R has at least one accumulation point in R.

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Theorem 1:

Given a function f and a point z0, suppose that:
a) f is analytic at z0
b) f(z0) = 0 but f(z) is not identically equal to zero in any neighborhood of z0.

Then f(z) does not equal 0 throughout some deleted neighborhood 0<|z - z0|< epsilon of z0.

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Using the Bolzano-Weierstrass Theorem and Theorem 1, prove that if a function f is analytic in a region R consisting of all points inside and on a simple closed contour C (except possibly poles inside C) and if all the zeros of f in R are interior to C and are of finite order, then those zeros must be finite in number.

Solution Summary

A complete, detailed proof is provided in the attached .pdf file.

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