State and prove Jordan's lemma.

What is the residue of a function $f$ at an isolated singularity $a$ ? If $f(z)=\frac{g(z)}{(z-a)^{k}}$ with $k$ a positive integer, $g$ analytic, and $g(a) \neq 0$, derive a formula for the residue of $f$ at $a$ in terms of derivatives of $g$.

Evaluate

$$\int_{-\infty}^{\infty} \frac{x^{3} \sin x}{\left(1+x^{2}\right)^{2}} d x$$