Question

Prove that an analytic function with a constant real part is a constant.

Answer 1

Answer:

If either Ref(z) = u or Imf(z) = v is constant, then f(z) is constant. In particular, a nonconstant analytic function cannot take only real or only pure imaginary values. ... If |f(z)| is constant or arg f(z) is constant, then f(z) is constant. For example, if f (z) = 0, then 0 = f (z) = ∂u ∂x + i ∂v ∂x .

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Answer 2

Answer:

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