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Part BCSIR NET December 2025check-which-poles-actually-lie-inside-each-radius-before-computing-any-residue

Check which poles actually lie inside each radius before computing any residue

Let γR(t)=2+i+Re(2π\gamma_R(t) = 2 + i + Re^(2\piit) for t ∈ [0,1] and R = 1, 2. Which of the following statements is true?

  1. A.∫γ1\int_{\gamma1} tan(z)dz =2πi= 2\pi{}i
  2. B.∫γ1\int_{\gamma1} tan(z)dz =−2πi= -2\pi{}i
  3. C.∫γ2\int_{\gamma2} tan(z)dz =2πi= 2\pi{}i
  4. D.∫γ2\int_{\gamma2} tan(z)dz =−2πi= -2\pi{}i✓

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.

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50 are analysed free — try those first.

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

More on this topic

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 . Found a mistake? Tell us — corrections are the fastest thing we act on.

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