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.© BrainMass Inc. brainmass.com December 24, 2021, 5:28 pm ad1c9bdddf
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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
There are two proofs regarding analytic functions in this solution.