(a) Using contour integration, evaluate : $\int_{-\infty}^{\infty} \f…

Question

(a) Using contour integration, evaluate : $\int_{-\infty}^{\infty} \frac{(x^2 - x + 2)dx}{x^4 + 10x^2 + 9}$

(b) Expand $f(z) = \frac{1}{z^2 \sinh z}$ as a Laurent's series about $z = 0$ and hence evaluate $\oint_C \frac{dz}{z^2 \sinh z}$ around the unit circle C.

(c) State and prove necessary conditions for a function $f(z)$ to be analytic in a region.