# Analytic Function Proofs and Real Functions

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1) Show that the real part of the function z^(1/2) is always positive.

2) Suppose f: G --> C ( C complex plane) is analytic and that G is connected. Show that if f(z) is real for all z in G, then f is a constant.

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#### Solution Preview

Please see the attached file.

analytic functions

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1). Show that the real part of the function z^(1/2) is always positive.

Proof. By definition, we know that

So, the real part of the function z^(1/2) is

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#### Solution Summary

There are two proofs regarding analytic functions in this solution.

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