Residue Theorem

An analytic function  whose Laurent series is given by

(1)

can be integrated term by term using a closed contour  encircling ,

			(2)
			(3)

The Cauchy integral theorem requires that the first and last terms vanish, so we have

(4)

where  is the complex residue. Using the contour  gives

(5)

so we have

(6)

If the contour  encloses multiple poles, then the theorem gives the general result

(7)

where  is the set of poles contained inside the contour. This amazing theorem therefore says that the value of a contour integral for any contour in the complex plane depends only on the properties of a few very special points insidethe contour.

The diagram above shows an example of the residue theorem applied to the illustrated contour  and the function

(8)

Only the poles at 1 and  are contained in the contour, which have residues of 0 and 2, respectively. The values of the contour integral is therefore given by

(9)

REFERENCES:

Knopp, K. "The Residue Theorem." §33 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 129-134, 1996.

Krantz, S. G. "The Residue Theorem." §4.4.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 48-49, 1999.