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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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) pp. Chapt. 4 (Translated from Russian) {{MR|0444912}} {{ZBL|0357.30002}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''1–2''' , Moscow (1976) pp. Vol. 1, Chapt. 2; Vol. 2, Chapt. 1 (In Russian) {{MR|}} {{ZBL|0799.32001}} {{ZBL|0732.32001}} {{ZBL|0732.30001}} {{ZBL|0578.32001}} {{ZBL|0574.30001}} </TD></TR></table>

====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|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. [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H.C. Hutchins, "Examples of commutative rings" , Polygonal (1981) {{MR|0638720}} {{ZBL|0492.13001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.M. Cohn, "Skew field constructions" , Cambridge Univ. Press (1977) {{MR|0463237}} {{ZBL|0355.16009}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. 241 {{MR|0510197}} {{MR|1535085}} {{MR|0188405}} {{MR|1570643}} {{MR|1528598}} {{MR|0054016}} {{ZBL|0395.30001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) {{MR|0344507}} {{ZBL|0271.32001}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979) {{MR|0593142}} {{MR|0197687}} {{MR|1523319}} {{ZBL|0477.30001}} {{ZBL|0336.30001}} {{ZBL|0005.21004}} {{ZBL|65.0302.01}} {{ZBL|58.0297.01}} </TD></TR></table>