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Compute $\int_{|z|=2} \frac{dz}{z^2-1}$ for the positive sense of the circle.

Please don't use knowledge after Cauchy Theorem and Cauchy Integration Formula.Thank you!

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    substitute $z=2e^{i\phi}$ and integrate from $\phi=0$ to $2\pi$. Fairly straight forward2012-04-09

1 Answers 1

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Substituting $z=2e^{i\phi}$ results in (using $dz=2i e^{i\phi}\, d\phi$) $ \int_{|z|=2} \frac{dz}{z^2-1} = \int_0^{2\pi}\frac{2 i e^{i\phi}}{4e^{2i\phi}-1}\,d\phi $ $ = \int_0^{\pi}\frac{2 i e^{i\phi}}{4e^{2i\phi}-1}\,d\phi + \int_{\pi}^{2\pi}\frac{2 i e^{i\phi}}{4e^{2i\phi}-1}\,d\phi $ $ = \int_0^{\pi}\frac{2 i e^{i\phi}}{4e^{2i\phi}-1}\,d\phi + \int_{0}^{\pi}\frac{2 i e^{i(\phi+\pi)}}{4e^{2i(\phi+\pi)}-1}\,d\phi $ $ = \int_0^{\pi}\frac{2 i e^{i\phi}}{4e^{2i\phi}-1}\,d\phi + \int_{0}^{\pi}\frac{-2 i e^{i\phi}}{4e^{2i\phi}-1}\,d\phi $ $ = 0 $

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    Ah, I hoped that it's such... thanks (+1 finally)2012-04-09