Part CCSIR NET December 2025z-sin-z-is-even-so-its-reciprocals-laurent-series-has-no-odd-powers-the-residue-at-the-double-pole-is-simply-0

Z sin z is even so its reciprocals laurent series has no odd powers the residue at the double pole is simply 0

For $t\in[0,2\pi]$ , let $\gamma_{1}(t)=e^($ it) and $\gamma_{2}(t)=1+i+e^($ it). Which of the following statements are true?

A. $\int_\gamma1$ dz/(z sin z) = 0✓
B. $\int_\gamma1$ dz $/(z \sin z) = 2\pi{}i$
C. $\int_\gamma2$ dz $/(z \sin z) = 2\pi{}i$
D. $\int_\gamma2$ dz/(z sin z) = 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.

50 are analysed free — try those first.

Related counterexample: ∮γ f = 0 implies f is holomorphic inside γ

More on this topic

residue of exp over powerDecember 2023
the two hypotheses fix the order of vanishingDecember 2024
on the unit circle conjugate is reciprocalDecember 2024
residues can cancelJune 2024
the residue at an essential singularity convolves both seriesJune 2025
a simple pole forces a nonzero residue take the contrapositive on the singleton claimDecember 2025

The chapter behind this: Residue theorem — the five standard patterns — free to read

From Singularities and Residues › Residue theorem and standard contour integrals

Last revised 10 Sept 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.

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