(a) State and prove Liouville's theorem. (b) Using calculus of residu…

Question

(a) State and prove Liouville's theorem.

(b) Using calculus of residues, evaluate : $\int_0^\infty \frac{\sin mx}{x} dx$ for $m > 0$.

(c) Determine the analytic function $f(z) = u + iv$, where $v = \frac{\sinh 2y}{\cos 2x + \cosh 2y}$.