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Part BCSIR NET June 2024residues-can-cancel

Residues can cancel

Let a, b be two real numbers such that a < 0 < b. For a positive real number r, define γr(t)=\gamma_r(t) = re^(it) (where t∈[0,2π])t \in [0, 2\pi]) and Ir=(1/2πi)∫(γr)(z2+1)/((z−a)(z−b))I_r = (1/2\pi{}i) \int_(\gamma_r) (z^{2} + 1) / ((z - a)(z - b)) dz. Which of the following statements is necessarily true?

  1. A.IrI_{r} ≠ 0 if r > max{|a|, b}
  2. B.IrI_{r} ≠ 0 if r < max{|a|, b}
  3. C.IrI_{r} = 0 if r > max{|a|, b} and |a| = b✓ Correct
  4. D.IrI_{r} = 0 if |a| < r < b

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 execution slip.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

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

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