A contour integral around a branch point

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This is the exact same question from the book 'Complex Variables and Applications'. Lucky I stumbled on this video because I was struggling with solving it. Thanks!

slog
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Hello MatheMagician, nice video. It is also interesting to observe what happens to the term π/4 (1-a)/cos(a π/2) when a approaches 1. The formula still holds, and the integral value is 1/2.

amoli
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I feel like there's something not quite right when you derivate log(z) to 1/z

Nightfold