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Part CCSIR NET June 2024analytic-nonvanishing-factor-cannot-remove

Analytic nonvanishing factor cannot remove

For z∈Cz \in \mathbb{C} \ {0}, let f(z) = (1/z)sin(1/z) and g(z) = f(z)sin(z). Which of the following statements are true?

  1. A.f has an essential singularity at 0✓ Correct
  2. B.g has an essential singularity at 0✓ Correct
  3. C.f has a removable singularity at 0
  4. D.g has a removable singularity at 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, and why this one tests standard counterexample.

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50 are analysed free — 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 ⇒ 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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