GATE 2025 MA – Question 63
Distinct nonzero real α,β and a polynomial Q of degree <5 define $f(z)=[\alpha^6\sin(\beta z)-\beta^6(e^{2\alpha z}-Q(z))]/z^6$. The source says f “satisfies Morera’s theorem in ℂ∖{0}”. Find α/(4β) (integer).
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Show answer and explanation
Correct answer: 8 to 8
Explanation
The official answer uses a zero integral around the puncture, equivalently zero residue. Since deg Q<5, it cannot affect the numerator’s z⁵ coefficient. That coefficient is $[\alpha^6\beta^5-\beta^6(2\alpha)^5]/120=\alpha^5\beta^5(\alpha-32\beta)/120$. Setting it to zero gives α=32β and **α/(4β)=8**. The wording is insufficient literally: f is already holomorphic on the punctured plane for every supplied α,β,Q, so the local Morera condition alone does not force this ratio. The official answer requires the additional global zero-period/residue interpretation.