Question Number 214279 by issac last updated on 03/Dec/24
$$\mathrm{evaluate} \\ $$$$\rho=\frac{\oint_{\:\mathcal{C}} \:\frac{{z}}{\mathrm{2sin}\left({z}\right)−\mathrm{2}{z}\mathrm{cos}\left({z}\right)−\pi}\:\mathrm{d}{z}}{\oint_{\:\mathcal{C}} \:\frac{\mathrm{2}}{\mathrm{2sin}\left({z}\right)−\mathrm{2}{z}\mathrm{cos}\left({z}\right)−\pi}\mathrm{d}{z}} \\ $$$$\mathcal{C}=\mid{z}−\frac{\mathrm{3}\pi}{\mathrm{4}}\mid=\frac{\pi}{\mathrm{4}} \\ $$