Contour Integration

Contour integration is the process of calculating the values of a contour integral around a given contour in the complex plane. As a result of a truly amazing property of holomorphic functions, such integrals can be computed easily simply by summing the values of the complex residues inside the contour.

Let and be polynomials of polynomial degree and with coefficients , ..., and , ..., . Take the contour in the upper half-plane, replace by , and write . Then

(1)

Define a path which is straight along the real axis from to and make a circular half-arc to connect the two ends in the upper half of the complex plane. The residue theorem then gives

			(2)
			(3)
			(4)

where denotes the complex residues. Solving,

(5)

Define

			(6)
			(7)
			(8)
			(9)

and set

(10)

then equation (9) becomes

(11)

Now,

(12)

for . That means that for , or , , so

(13)

for . Apply Jordan's lemma with . We must have

(14)

so we require .

Then

(15)

for and . Since this must hold separately for real and imaginary parts, this result can be extended to

(16)

(17)