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Show that $$ \int_{-1}^{1} \frac{\log|z-x|}{\pi\sqrt{1-x^2}}dx = \log{\frac{|z+\sqrt{z^2-1}|}{2}},\quad z \in \mathbb{C} $$

How can I apply the Joukowski conformal map to this problem? Thanks.

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A trivial question: what is the motivation behind this integral? Are you expecting a contour integral or something to solve this? It does not seem to related to conformal mapping. – Kerry Mar 10 '12 at 6:15

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