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Part BCSIR NET December 2023

Cauchy–Riemann equations, harmonic functions

Let ℍ = {z∈C:Imz>0z \in \mathbb{C} : Im z > 0} and f(z) = eize^{iz}. Which one of the following statements is true?

  1. A.f(ℍ)=C) = \mathbb{C}∖{0}.
  2. B.f(ℍ) ∩ ℍ is countable.
  3. C.f(ℍ) is bounded.✓ Correct
  4. D.f(ℍ) is a convex subset of C\mathbb{C}.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

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Related counterexample: A harmonic function on any domain has a harmonic conjugate

More on this topic

The chapter behind this: Cauchy–Riemann — what they do and do not give you — free to read

From Analytic Functions › Cauchy–Riemann equations, harmonic functions

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