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

No holomorphic function decays that fast

Let disc 𝔻 = { : |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) = 0✓
  2. B.f(z) = 0 for all z ∈ 𝔻.✓
  3. C.There exists a nonzero constant c such that f(z) = ce^(−1/z) for all z ∈ 𝔻 \ {0}.
  4. D.There exists a nonzero constant c and a positive integer n such that f(z) = cz 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 — the specific reasoning error each wrong option was built to reward.

See pricing

50 are analysed free — try those first.

Related counterexample: |f| bounded near an isolated singularity ⇒ pole

More on this topic

The chapter behind this: Singularities — three types, three tests — free to read

From Singularities and Residues › Laurent series, classification of singularities, Casorati–Weierstrass

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