Computing integrals for trigonometric-polynomial quotients

Description

Here, we describe the integration procedure for integrals of the form:

$\operatorname {PV} \int _{-\infty }^{\infty }{\frac {P(x)\sin x}{Q(x)}}$

The $\sin$ may be replaced by $\cos$.

Here, we assume that the polynomial $Q$ does not have any zeros on the real axis (except possibly at the origin).

Solution by semicircular contour integration

Further information: semicircular contour integration method

This is typical, for instance, when the integrand is of the form:

$f(x):={\frac {\sin x}{Q(x)}}$

where the degree of $Q$ is strictly greater than one, and $Q$ has no real zeros. We use the complex-valued function $e^{iz}/Q(z)$. In this case, Jordan's lemma (or the weaker version, the so-called [semicircular contour theorem]]) tells us that the integral along large semicircles tends to zero, so:

$\operatorname {PV} \int _{-\infty }^{\infty }{\frac {\sin x}{Q(x)}}=\operatorname {Im} \left((2\pi i)\sum \operatorname {res} (f;z_{j})\right)$

where the sum is taken over all the zeros $z_{j}$ of $f$ that lie in the upper half-plane.

Note that we can drop the $\operatorname {PV}$ (Cauchy principal value symbol), since the integral is absolutely convergent in this case. Also,if $Q$ has only simple poles, so that we have:

$Q(z):=\prod _{j=1}^{n}(z-z_{j})$

Then, we get the formula:

$\int _{-\infty }^{\infty }{\frac {\sin x}{Q(x)}}=2\pi \operatorname {Re} \left(\sum _{j=1}^{n}{\frac {\sin(z_{j})}{\prod _{k\neq j}(z_{j}-z_{k})}}\right)$