Laurent series

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.

References

[1]	A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) pp. Chapt. 4 (Translated from Russian) MR0444912 Zbl 0357.30002
[2]	B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) pp. Vol. 1, Chapt. 2; Vol. 2, Chapt. 1 (In Russian) Zbl 0799.32001 Zbl 0732.32001 Zbl 0732.30001 Zbl 0578.32001 Zbl 0574.30001

Comments

Let be any field. The term Laurent series is also often used to denote a formal expansion of the form

The expressions are added termwise and multiplied as follows:

where

(note that this sum is finite). The result is a field, denoted by . It is the quotient field of the ring of formal power series , and is called the field of formal Laurent series. A valuation is defined by if . This makes a discretely valued complete field; the ring of integers is , the maximal ideal is and the residue field is . (Cf. also Valuation.)

A Laurent polynomial over is an expression , , .

More generally one also defines (formal) Laurent series in several variables and non-commutative Laurent series, cf. [a1].

References

[a1]	H.C. Hutchins, "Examples of commutative rings" , Polygonal (1981) MR0638720 Zbl 0492.13001
[a2]	P.M. Cohn, "Skew field constructions" , Cambridge Univ. Press (1977) MR0463237 Zbl 0355.16009
[a3]	L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. 241 MR0510197 MR1535085 MR0188405 MR1570643 MR1528598 MR0054016 Zbl 0395.30001
[a4]	L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) MR0344507 Zbl 0271.32001
[a5]	E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979) MR0593142 MR0197687 MR1523319 Zbl 0477.30001 Zbl 0336.30001 Zbl 0005.21004 Zbl 65.0302.01 Zbl 58.0297.01