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Part CCSIR NET June 2023conjugation-and-analyticity

Conjugation and analyticity

Let f(z) be an entire function on C\mathbb{C}. Which of the following statements are true?

  1. A.f(z̄) is an entire function.
  2. B.conj(f(z)) is an entire function.
  3. C.conj(f(z̄)) is an entire function.✓ Correct
  4. D.conj(f(z̄)) + f(z̄) is an entire function.

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, and why this one tests base field or ring.

See pricing

50 are analysed free — try those first.

The trap it tests

Base field or ring

The answer changes with the field or ring you are working over.

Drill statements like this

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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