Integrating Quotients of Polynomials Along the Real Axis

Ifwhereandare polynomials with the degree ofat least two more than the degree ofandhas only simple, non real poles then we can findusing the contour integral below.

In the diagram aboveis the semicircular contour of radiusBy Cauchy's integral formula

where theare the simple poles ofin the upper half plane.

By Jordan's Lemmaso

Example: Find

Consider the integral(The two integrals have the same value since both are evaluated along the real axis).

The degree of the denominator is at least two more than the degree of the numerator so we can use Jordan's Lemma.

The zeroes ofare are

andare in the upper half plane so we evaluateat these points.


We can simplify this: