If is analytic throughout the annular region between and on the concentric circles and centered at and of radii and respectively, then there exists a unique series expansion in terms of positive and negative powers of ,

(1)

where

			(2)
			(3)

(Korn and Korn 1968, pp. 197-198).

Let there be two circular contours and , with the radius of larger than that of . Let be at the center of and , and be between and . Now create a cut line between and , and integrate around the path , so that the plus and minus contributions of cancel one another, as illustrated above. From the Cauchy integral formula,

			(4)
			(5)
			(6)

Now, since contributions from the cut line in opposite directions cancel out,

			(7)
			(8)
			(9)

For the first integral, . For the second, . Now use the Taylor series (valid for )

(10)

to obtain

			(11)
			(12)
			(13)

where the second term has been re-indexed. Re-indexing again,

(14)

Since the integrands, including the function , are analytic in the annular region defined by and , the integrals are independent of the path of integration in that region. If we replace paths of integration and by a circle of radius with , then

			(15)
			(16)
			(17)

Generally, the path of integration can be any path that lies in the annular region and encircles once in the positive (counterclockwise) direction.

The complex residues are therefore defined by

(18)

Note that the annular region itself can be expanded by increasing and decreasing until singularities of that lie just outside or just inside are reached. If has no singularities inside , then all the terms in (◇) equal zero and the Laurent series of (◇) reduces to a Taylor series with coefficients .