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Part CCSIR NET June 2023conjugation-and-analyticity

Conjugation and analyticity

Let f(z) be an entire function on . 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.✓
  4. D.conj(f(z̄)) + f(z̄) is an entire function.

Solution

If then conj ā is entire. z ↦ f(z̄) and z ↦ conj(f(z)) are anti-holomorphic (entire only if f is constant). (4) is entire plus non-entire ⇒ not entire in general.

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

From Analytic Functions › Cauchy–Riemann equations, harmonic functions

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