# Laurent series

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A generalization of a power series in non-negative integral powers of the difference or in non-positive integral powers of in the form (1)

The series (1) is understood as the sum of two series: the regular part of the Laurent series, and the principal part of the Laurent series. The series (1) is assumed to converge if and only if its regular and principal parts converge. Properties of Laurent series: 1) if the domain of convergence of a Laurent series contains interior points, then this domain is a circular annulus with centre at the point ; 2) at all interior points of the annulus of convergence the series (1) converges absolutely; 3) as for power series, the behaviour of a Laurent series at points on the bounding circles and can be very diverse; 4) on any compact set the series (1) converges uniformly; 5) the sum of the series (1) in is an analytic function ; 6) the series (1) can be differentiated and integrated in term-by-term; 7) the coefficients of a Laurent series are defined in terms of its sum by the formulas (2)

where is any circle with centre situated in ; and 8) expansion in a Laurent series is unique, that is, if in , then all the coefficients of their Laurent series in powers of coincide.

For the case of a centre at the point at infinity, , the Laurent series takes the form (3)

and now the regular part is while the principal part is The domain of convergence of (3) has the form and formulas (2) go into where . Otherwise all the properties are the same as in the case of a finite centre .

The application of Laurent series is based mainly on Laurent's theorem (1843): Any single-valued analytic function in an annulus can be represented in by a convergent Laurent series (1). In particular, in a punctured neighbourhood of an isolated singular point of single-valued character an analytic function can be represented by a Laurent series, which serves as the main instrument for investigating its behaviour in a neighbourhood of an isolated singular point.

For holomorphic functions of several complex variables the following proposition can be regarded as the analogue of Laurent's theorem: Any function , holomorphic in the product of annuli , can be represented in as a convergent multiple Laurent series (4)

is which the summation extends over all integral multi-indices   where is the product of the circles , . The domain of convergence of the series (4) is logarithmically convex and is a relatively-complete Reinhardt domain. However, the use of multiple Laurent series (4) is limited, since for holomorphic functions cannot have isolated singularities.