# Application residue theorem for improper integrals

Let $R(z)=\displaystyle \frac{P(z)}{Q(z)}$ be a rational function of order(Q) $\geq \mathrm{order}(P)+2$ and $Q(x)\neq 0$ for all $x\in \mathbb{R}$. Then we have: $$\int_{-\infty}^{\infty}R(x)\mathrm{d}x=2\pi i\sum_{z:\ \mathrm{Im} \ z>0}{\rm Res}(R,z)$$

Why is

order(Q) $\geq \mathrm{order}(P)+2$ and $Q(x)\neq 0$ for all $x\in \mathbb{R}$

important?

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I guess you meant "Let $\,R(z)\,$ be a rational function with rational coefficients..." , i.e. $\,P(z), Q(z)\in\mathbb{Q}[z]\,$ ....? –  DonAntonio Jun 6 '12 at 23:46
I fixed it. The coefficients are complex. –  Chris Jun 7 '12 at 0:15