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 inside the 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)

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