Paper 1, Section I, 2D2 \mathrm{D}

Complex Analysis or Complex Methods
Part IB, 2013

Classify the singularities (in the finite complex plane) of the following functions: (i) 1(coshz)2\frac{1}{(\cosh z)^{2}}; (ii) 1cos(1/z)\frac{1}{\cos (1 / z)}; (iii) 1logz(π<argz<π)\frac{1}{\log z} \quad(-\pi<\arg z<\pi); (iv) z121sinπz(π<argz<π)\frac{z^{\frac{1}{2}}-1}{\sin \pi z} \quad(-\pi<\arg z<\pi).