(a) Find the nature of singularities and residue at these points of t…

Previous year question for Mathematics-III.

Question

(a) Find the nature of singularities and residue at these points of the function $f(z) = \frac{e^z}{(z^2 + \pi^2)}$.

(b) State Residue theorem and evaluate the following integral using the Residue theorem

$$\oint_{C} \frac{1 - 3z}{(z-1)(z-2)(z+4)} \, dz$$

where $C$ is the circle $|z| = 3$.