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)

REFERENCES:

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 406-409, 1985.

Krantz, S. G. "Applications to the Calculation of Definite Integrals and Sums." §4.5 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 51-63, 1999.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 353-356, 1953.

Whittaker, E. T. and Watson, G. N. "The Evaluation of Certain Types of Integrals Taken Between the Limits  and ," "Certain Infinite Integrals Involving Sines and Cosines," and "Jordan's Lemma." §6.22-6.222 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 113-117, 1990.