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Part CCSIR NET December 2024no-holomorphic-function-decays-that-fast

No holomorphic function decays that fast

Let disc 𝔻 = {z∈Cz \in \mathbb{C} : |z| < 1} and f be a holomorphic function on 𝔻 such that the function g(z) = e^(1/z)f(z) on 𝔻 \ {0} is bounded. Which of the following statements are true?

  1. A.f(0) = 0βœ“ Correct
  2. B.f(z) = 0 for all z ∈ 𝔻.βœ“ Correct
  3. C.There exists a nonzero constant c such that f(z) = ce^(βˆ’1/z) for all z ∈ 𝔻 \ {0}.
  4. D.There exists a nonzero constant c and a positive integer n such that f(z) = czne(βˆ’1/z)^{n}e^(-1/z) for all z ∈ 𝔻 \ {0}.

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

Not a worked solution repeated four times β€” the specific reasoning error each wrong option was built to reward, and why this one tests standard counterexample.

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50 are analysed free β€” try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

Related counterexample: |f| bounded near an isolated singularity β‡’ pole

More on this topic

The chapter behind this: Singularities β€” three types, three tests β€” free to read

From Singularities and Residues β€Ί Laurent series, classification of singularities, Casorati–Weierstrass

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

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