Contour Integration

Contour integration is the process of evaluating a contour integral along a contour in the complex plane. For a positively oriented simple closed contour, the residue theorem expresses the contour integral of a meromorphic function as times the sum of its complex residues inside the contour, provided it has no poles on the contour.

Let and be polynomials of polynomial degree and , respectively, and suppose has no zeros on the real axis. For , the improper integral is given by

(1)

Let consist of the segment from to along the real axis, followed by the semicircle , , in the upper half-plane. For large enough to include all poles in the upper half-plane, the residue theorem gives

			(2)
			(3)

where denotes the complex residue at each indicated pole. Solving for the integral along the real axis gives

(4)

where

(5)

The triangle inequality and the leading terms of the polynomials give

(6)

so as . Consequently,

(7)

For and , Jordan's lemma makes the semicircle contribution vanish for the oscillatory integral, yielding

(8)

If and have real coefficients, taking the real part and imaginary part gives

(9)

(10)