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Part BCSIR NET December 2024orders-subtract-so-the-pole-cancels

Orders subtract so the pole cancels

Consider the entire function ᶻ − e⁻ᶻ) and the meromorphic function ᶻ − e⁻ᶻ) on . Which of the following statements is true?

  1. A.z = 0 is a zero of f of order 3 and a pole of g of order 1.
  2. B.z = 0 is a zero of f of order 2 and a pole of g of order 1.
  3. C.z = 0 is a zero of f of order 3 and a zero of g of order 1.✓
  4. D.z = 0 is a zero of f of order 2 and a zero of g of order 1.

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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