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Evaluate a line integral

Short answer

Set z-a = e^(iθ). Then dz = i e^(iθ) dθ, so the integral becomes ∫_0^{2π} i dθ = 2πi, consistent with Cauchy's integral formula for f(z)=1.

Show that ∮_C dz/(z − a) = 2πi where C is the unit circle |z − a| = 1.

On the circle |z − a| = 1, put z − a = e^(iθ), so z = a + e^(iθ), 0 ≤ θ ≤ 2π. Then dz = i e^(iθ) dθ.

∮_C dz/(z − a) = ∫₀^2π (i e^(iθ) dθ) / e^(iθ)
               = i ∫₀^2π dθ
               = 2πi

Hence ∮_C dz/(z − a) = 2πi. (This also follows from Cauchy's integral formula with f(z) = 1, as the point a lies inside C.)

✓ Answer checked by Claude Opus 5.5 (high effort)

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