Laurent series

A generalization of a power series in non-negative integral powers of the difference $ z - a $ or in non-positive integral powers of $ z - a $ in the form

$$ \tag{1 } \sum _ {k = - \infty } ^ {+\infty } c _ {k} ( z - a ) ^ {k} . $$

The series (1) is understood as the sum of two series:

$$ \sum_{k=0}^ {+ \infty } c _ {k} ( z - a ) ^ {k} , $$

the regular part of the Laurent series, and

$$ \sum_{k=-\infty}^{-1} c_k ( z - a ) ^ {k} , $$

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 $ D = \{ {z \in \mathbf C } : {0 \leq r < | z - a | < R \leq + \infty } \} $ with centre at the point $ a \neq \infty $; 2) at all interior points of the annulus of convergence $ D $ the series (1) converges absolutely; 3) as for power series, the behaviour of a Laurent series at points on the bounding circles $ | z- a | = r $ and $ | z - a | = R $ can be very diverse; 4) on any compact set $ K \subset D $ the series (1) converges uniformly; 5) the sum of the series (1) in $ D $ is an analytic function $ f ( z) $; 6) the series (1) can be differentiated and integrated in $ D $ term-by-term; 7) the coefficients $ c _ {k} $ of a Laurent series are defined in terms of its sum $ f ( z) $ by the formulas

$$ \tag{2 } c _ {k} = \frac{1}{2 \pi i } \int\limits _ \gamma \frac{f ( z) d z }{( z - a ) ^ {k+1} } ,\ \ k = 0 , \pm 1 \dots $$

where $ \gamma = \{ {z } : {| z - a | = \rho, r < \rho < R } \} $ is any circle with centre $ a $ situated in $ D $; and 8) expansion in a Laurent series is unique, that is, if $ f ( z) \equiv \phi ( z) $ in $ D $, then all the coefficients of their Laurent series in powers of $ z - a $ coincide.

For the case of a centre at the point at infinity, $ a = \infty $, the Laurent series takes the form

$$ \tag{3 } \sum _ {k = - \infty } ^ { + \infty } c _ {k} z ^ {k} , $$

and now the regular part is

$$ \sum _ {k = - \infty } ^ { 0 } c _ {k} z ^ {k} , $$

while the principal part is

$$ \sum _ {k = 1 } ^ { {+ } \infty } c _ {k} z ^ {k} . $$

The domain of convergence of (3) has the form

$$ D ^ \prime = \{ {z } : {0 \leq r < | z | < R \leq + \infty } \} , $$

and formulas (2) go into

$$ c _ {k} = - \frac{1}{2 \pi i } \int\limits _ \gamma z ^ {k+1} f ( z) d z ,\ \ k = 0 , \pm 1 \dots $$

where $ \gamma = \{ {z } : {| z | = \rho, r < \rho < R } \} $. Otherwise all the properties are the same as in the case of a finite centre $ a $.

The application of Laurent series is based mainly on Laurent's theorem (1843): Any single-valued analytic function $ f ( z) $ in an annulus $ D = \{ {z } : {0 \leq r < | z- a | < R \leq + \infty } \} $ can be represented in $ D $ by a convergent Laurent series (1). In particular, in a punctured neighbourhood $ D = \{ {z } : {0 < | z - a | < R } \} $ of an isolated singular point $ a $ of single-valued character an analytic function $ f ( z) $ 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 $ f ( z) $ of several complex variables $ z = ( z _ {1} \dots z _ {n} ) $ the following proposition can be regarded as the analogue of Laurent's theorem: Any function $ f ( z) $, holomorphic in the product $ D $ of annuli $ D _ \nu = \{ {z _ \nu \in \mathbf C } : {0 \leq r _ \nu < | z _ \nu - a _ \nu | < R _ \nu \leq + \infty } \} $, can be represented in $ D $ as a convergent multiple Laurent series

$$ \tag{4 } f ( z) = \sum _ {| k| = - \infty } ^ { {+ } \infty } c _ {k} ( z - a ) ^ {k} , $$

is which the summation extends over all integral multi-indices

$$ k = ( k _ {1} \dots k _ {n} ) ,\ \ | k | = k _ {1} + \dots + k _ {n} , $$

$$ ( z - a ) ^ {k} = ( z _ {1} - a _ {1} ) ^ {k _ {1} } \dots ( z _ {n} - a _ {n} ) ^ {k _ {n} } , $$

$$ c _ {k} = \frac{1}{( 2 \pi i ) ^ {n} } \int\limits _ \gamma \frac{f ( z) d z }{( z - a ) ^ {k+n}} , $$

where $ \gamma $ is the product of the circles $ \gamma _ \nu = \{ {z _ \nu \in \mathbf C } : {z _ \nu = a _ \nu + \rho _ \nu e ^ {it} , r _ \nu < \rho _ \nu < R _ {\nu, } 0 \leq t \leq 2 \pi } \} $, $ \nu = 1 \dots n $. 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 $ n \geq 2 $ holomorphic functions $ f ( z) $ 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 $ k $ be any field. The term Laurent series is also often used to denote a formal expansion of the form

$$ \sum _ {i= - N } ^ \infty a _ {i} X ^ {i} ,\ \ a _ {i} \in k ,\ N \in \mathbf Z . $$

The expressions are added termwise and multiplied as follows:

$$ \left ( \sum _ {i= - N } ^ \infty a _ {i} X ^ {i} \right ) \left ( \sum _ {i= - M } ^ \infty b _ {i} X ^ {i} \right ) = \ \sum _ {j= - N - M } ^ \infty c _ {j} X ^ {j} , $$

where

$$ c _ {j} = \sum _ {\begin{array}{c} k+ l = j \\ k,l \in \mathbf Z \end{array} } a _ {k} b _ {l} $$

(note that this sum is finite). The result is a field, denoted by $ k(( X)) $. It is the quotient field of the ring of formal power series $ k [[ X ]] $, and is called the field of formal Laurent series. A valuation is defined by $ v ( \sum _ {i= - N } ^ \infty a _ {i} X ^ {i} ) = - N $ if $ a _ {-N} \neq 0 $. This makes $ k(( X)) $ a discretely valued complete field; the ring of integers is $ k[[ X ]] $, the maximal ideal is $ X k [[ X]] $ and the residue field is $ k $. (Cf. also Valuation.)

A Laurent polynomial over $ k $ is an expression $ \sum _ {i = - N } ^ {M} a _ {i} X ^ {i} $, $ - N \leq M $, $ N, M \in \mathbf Z $.

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