Ifwhere
and
are polynomials with the degree of
at least two more than the degree of
and
has only simple, non real poles then we can find
using the contour integral below.
In the diagram aboveis the semicircular contour of radius
By Cauchy's integral formula
where the
are the simple poles of
in 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
and
are in the upper half plane so we evaluate
at these points.
and
We can simplify this:
Hence