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Which of the following statements involving contour integrals (evaluated counter-clockwise) on the unit circle C in the complex plane is/are TRUE?

1. ∮_C e^z dz = 0
2. ∮_C zⁿ dz = 0, where n is an even integer
3. ∮_C cos z dz ≠ 0
4. ∮_C sec z dz ≠ 0

Correct answer: ∮_C e^z dz = 0

Solution

The integral of the function e^z over a closed contour, such as the unit circle, is zero due to Cauchy's integral theorem, which states that the integral of an analytic function over a closed curve is zero if the function is analytic inside the curve.

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