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!
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!
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 $