Calculate $$\int_0^\pi\log\sin\theta\, d\theta$$ using contour integration only. I know the problem can be solved in multiple ways. But now I am interested in only the solution using contour integration. I have tried the usual techniques like setting $\sin\theta=\frac{e^{i\theta}-e^{i\theta}}{2i}$ but with no luck. And it doesn't really seem like other integrals that can be solved using contour integration. Any suggestions?
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1This may help: Showing that $\int_0^1 \log(\sin \pi x)dx=-\log2$. – Minus One-Twelfth Jun 24 '19 at 08:46
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Rudin's RCA does this under the heading 'Jensen's Formula'. – Kavi Rama Murthy Jun 24 '19 at 08:49
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1using the symmetry of the sine function this seems easy to solve but not using contour integration – Jun 24 '19 at 09:01