In this problem you are guided through the start of the proof of the proposition:
If u(x, y) is a harmonic function and v(x, y) is its harmonic conjugate, then the function φ(x, y) = u(x, y)v(x, y) is harmonic.
Proof Suppose f(z) = u(x, y)+iv(x, y) is analytic in a domain D. We saw that the product of two analytic functions is analytic. Hence [f(z)]2 is analytic. Now examine [f(z)]2 and finish the proof.
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