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Part CCSIR NET December 2024only-multiples-of-four-survive

Only multiples of four survive

Let f:C→Cf : \mathbb{C} \to \mathbb{C} be an entire function such that f(z) = f(iz) for all z∈Cz \in \mathbb{C}. Which of the following statements are true?

  1. A.f(z) = f(−z) for all z∈Cz \in \mathbb{C}.✓ Correct
  2. B.f′(0) = f″(0) = f‴(0) = 0✓ Correct
  3. C.There is an entire function g:C→Cg : \mathbb{C} \to \mathbb{C} such that f(z)=g(z4)f(z) = g(z^{4}) for all z∈Cz \in \mathbb{C}.✓ Correct
  4. D.f is necessarily a constant 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 execution slip.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

Related counterexample: Zeros of a non-constant holomorphic function cannot accumulate

More on this topic

The chapter behind this: Analyticity, the identity theorem and zeros — free to read

From Analytic Functions › Power series and analyticity

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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