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Part BCSIR NET June 2024parametrisation-winds-twice

Parametrisation winds twice

Consider the contour γ\gamma given by γ(θ)=e(2iθ)\gamma(\theta) = e^(2i\theta) for θ∈[0,π/2];γ(θ)=1+2e(2iθ)\theta \in [0, \pi/2]; \gamma(\theta) = 1 + 2e^(2i\theta) for θ∈[π/2,3π/2];γ(θ)=e(2iθ)\theta \in [\pi/2, 3\pi/2]; \gamma(\theta) = e^(2i\theta) for θ∈[3π/2,2π]\theta \in [3\pi/2, 2\pi]. Then what is the value of ∫γ\int_\gamma dz / (z(z − 2))?

  1. A.0
  2. B.πi\pi{}i
  3. C.−πi-\pi{}i✓ Correct
  4. D.2πi2\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, and why this one tests execution slip.

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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: Cauchy's theorem applies to any domain without singularities of f

More on this topic

The chapter behind this: Cauchy's theorem and integral formula — the toolkit — free to read

From Cauchy Theory › Cauchy's theorem and integral formula

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