A function $f(z)$ has an isolated singularity at $a$, with Laurent expansion

$$f(z)=\sum_{n=-\infty}^{\infty} c_{n}(z-a)^{n}$$

(a) Define res $(f, a)$, the residue of $f$ at the point $a$.

(b) Prove that if $a$ is a pole of order $k+1$, then

$$\operatorname{res}(f, a)=\lim _{z \rightarrow a} \frac{h^{(k)}(z)}{k !}, \quad \text { where } \quad h(z)=(z-a)^{k+1} f(z) .$$

(c) Using the residue theorem and the formula above show that

$$\int_{-\infty}^{\infty} \frac{d x}{\left(1+x^{2}\right)^{k+1}}=\pi \frac{(2 k) !}{(k !)^{2}} 4^{-k}, \quad k \geq 1$$