# 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

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
How to Cite This Entry:
Laurent series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Laurent_series&oldid=47592
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article