91Ó°ÊÓ

Identify the zeroes, poles and essential singularities of the following functions: (a) \(\tan z\) (b) \(\left[(z-2) / z^{2}\right] \sin [1 /(1-z)]\), (c) \(\exp (1 / z)\), (d) \(\tan (1 / z)\), (e) \(z^{2 / 3}\)

By applying the residue theorem around a wedge-shaped contour of angle \(2 \pi / n\), with one side along the real axis, prove that the integral $$ \int_{0}^{\infty} \frac{d x}{1+x^{n}} $$ where \(n\) is real and \(\geq 2\), has the value \((\pi / n) \operatorname{cosec}(\pi / n)\).

By considering the integral of $$ \left(\frac{\sin \alpha z}{\alpha z}\right)^{2} \frac{\pi}{\sin \pi z}, \quad \alpha<\frac{\pi}{2} $$ around a circle of large radius, prove that $$ \sum_{m=1}^{\infty}(-1)^{m-1} \frac{\sin ^{2} m \alpha}{(m \alpha)^{2}}=\frac{1}{2} $$