Skip to content
Part BCSIR NET December 2024on-the-unit-circle-conjugate-is-reciprocal

On the unit circle conjugate is reciprocal

For integers m, n ≥ 1, let I(m,n)=(1/2πi)∫CzmzˉnI_(m,n) = (1/2\pi{}i)\int_C z^{m} {\bar{z}}^{n} dz, where C is the circle {z∈Cz \in \mathbb{C} : |z| = 1} oriented counterclockwise. Which of the following statements is true?

  1. A.I_(m,n) = 1 if m = n
  2. B.I_(m,n) = 1 if m + 1 = n✓ Correct
  3. C.I_(m,n) = 1 if m = n + 1
  4. D.I_(m,n) = 1 if m = n + 2

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 standard counterexample.

See pricing

50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

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

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

ShareWhatsAppTelegram