Why do so many half-circle perimeters contribute $0$ to the answer in contour integration?

It could be that the phenomenon you ask about is an illusion; perhaps it "almost always" happens for the integrals that can be done this way, not for a random improper integral. A thoughtful calculus student thinks that most continuous functions are differentiable, while some do not have antiderivatives, when the reality is just the opposite.

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Integration

Complex Analysis

Contour Integration

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