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Suppose $f(z)$ is a analytic function inside and on the contour $|m|=1$, by using Residue Theorem, $\int_{|m|=\rho}\frac{f(z)}{z-1}dz=2\pi f(1)$ for any $\rho>1$, but how to calculate the integration $\int_{|m|=1}\frac{f(z)}{z-1}dz$? It seems that the answer is $\pi f(1)$, but how to prove it. Are there any reference?

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The magic words are "Cauchy principal value" –  Igor Rivin Nov 10 '11 at 11:15

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