Laurent series

In mathematics, the Laurent series of a complex function

is a representation of that function as a power series which includes terms of negative degree.

It may be used to express complex functions in cases where a Taylor series expansion cannot be applied.

Karl Weierstrass had previously described it in a paper written in 1841 but not published until 1894.

is counterclockwise around a Jordan curve enclosing

The annulus is shown in red in the figure on the right, along with an example of a suitable path of integration labeled

, this amounts to computing the complex Fourier coefficients of the restriction of

[4] The fact that these integrals are unchanged by a deformation of the contour

One may also obtain the Laurent series for a complex function

; instead, one often pieces together the Laurent series by combining known Taylor expansions.

Laurent series with complex coefficients are an important tool in complex analysis, especially to investigate the behavior of functions near singularities.

More generally, Laurent series can be used to express holomorphic functions defined on an annulus, much as power series are used to express holomorphic functions defined on a disc.

These radii can be computed by taking the limit superior of the coefficients

, there always exists a unique Laurent series with center

For example, consider the following rational function, along with its partial fraction expansion:

(which yields a power series) will only converge in a disc of radius 1, since it "hits" the singularity at

, where k is an arbitrary integer, and integrate on a path γ inside the annulus,

, where ε is a positive number small enough for γ to be contained in the constricted closed annulus, so the integration and summation can be interchanged.

of order equal to (negative) the degree of the highest term; on the other hand, if

If the inner radius of convergence of the Laurent series for

if and only if the principal part is an infinite sum, and has a pole otherwise.

may have infinitely many negative terms but still be regular at

, as in the example above, in which case it is represented by a different Laurent series in a disk about

, and can be analyzed similarly—while Laurent series with infinitely many negative terms have complicated behavior on the inner circle of convergence.

Algebraically, the expression for the terms of the product may involve infinite sums which need not converge (one cannot take the convolution of integer sequences).

Geometrically, the two Laurent series may have non-overlapping annuli of convergence.

Two Laurent series with only finitely many negative terms can be multiplied: algebraically, the sums are all finite; geometrically, these have poles at

Thus when defining formal Laurent series, one requires Laurent series with only finitely many negative terms.

Similarly, the sum of two convergent Laurent series need not converge, though it is always defined formally, but the sum of two bounded below Laurent series (or any Laurent series on a punctured disk) has a non-empty annulus of convergence.

, by the sum and multiplication defined above, formal Laurent series would form a field

A Laurent series is defined with respect to a particular point and a path of integration γ. The path of integration must lie in an annulus, indicated here by the red color, inside which is holomorphic ( analytic ).
e −1/ x 2 and its Laurent approximations (labeled) with the negative degree rising. The neighborhood around the zero singularity can never be approximated.
e −1/ x 2 and its Laurent approximations. As the negative degree of the Laurent series rises, it approaches the correct function.