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A generalization of a [[Power series|power series]] in non-negative integral powers of the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l0576901.png" /> or in non-positive integral powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l0576902.png" /> in the form
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l0576903.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
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 +
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 +
 
 +
A generalization of a [[Power series|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:
 
The series (1) is understood as the sum of two series:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l0576904.png" /></td> </tr></table>
+
$$
 +
\sum _ { k= } 0 ^ { {+ }  \infty }
 +
c _ {k} ( z - a )  ^ {k} ,
 +
$$
  
 
the regular part of the Laurent series, and
 
the regular part of the Laurent series, and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l0576905.png" /></td> </tr></table>
+
$$
 +
\sum _ {k = - \infty } ^ { - }  1
 +
c _ {k} ( z - a )  ^ {k} ,
 +
$$
 +
 
 +
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 }
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l0576906.png" /> with centre at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l0576907.png" />; 2) at all interior points of the annulus of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l0576908.png" /> the series (1) converges absolutely; 3) as for power series, the behaviour of a Laurent series at points on the bounding circles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l0576909.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769010.png" /> can be very diverse; 4) on any compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769011.png" /> the series (1) converges uniformly; 5) the sum of the series (1) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769012.png" /> is an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769013.png" />; 6) the series (1) can be differentiated and integrated in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769014.png" /> term-by-term; 7) the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769015.png" /> of a Laurent series are defined in terms of its sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769016.png" /> by the formulas
+
\int\limits _  \gamma
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\frac{f ( z)  d z }{( z - a )  ^ {k+} 1 }
 +
,\ \
 +
k = 0 , \pm  1 \dots
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769018.png" /> is any circle with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769019.png" /> situated in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769020.png" />; and 8) expansion in a Laurent series is unique, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769021.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769022.png" />, then all the coefficients of their Laurent series in powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769023.png" /> coincide.
+
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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769024.png" />, the Laurent series takes the form
+
For the case of a centre at the point at infinity, $  a = \infty $,  
 +
the Laurent series takes the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
\sum _ {k = - \infty } ^ { {+ }  \infty }
 +
c _ {k} z  ^ {k} ,
 +
$$
  
 
and now the regular part is
 
and now the regular part is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769026.png" /></td> </tr></table>
+
$$
 +
\sum _ {k = - \infty } ^ { 0 }
 +
c _ {k} z  ^ {k} ,
 +
$$
  
 
while the principal part is
 
while the principal part is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769027.png" /></td> </tr></table>
+
$$
 +
\sum _ {k = 1 } ^ { {+ }  \infty }
 +
c _ {k} z  ^ {k} .
 +
$$
  
 
The domain of convergence of (3) has the form
 
The domain of convergence of (3) has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769028.png" /></td> </tr></table>
+
$$
 +
D  ^  \prime  = \{ {z } : {0 \leq  r < | z | < R \leq  + \infty } \}
 +
,
 +
$$
  
 
and formulas (2) go into
 
and formulas (2) go into
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769029.png" /></td> </tr></table>
+
$$
 +
c _ {k}  = -  
 +
\frac{1}{2 \pi i }
 +
 
 +
\int\limits _  \gamma
 +
z  ^ {k+} 1 f ( z)  d z ,\ \
 +
k = 0 , \pm  1 \dots
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769030.png" />. Otherwise all the properties are the same as in the case of a finite centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769031.png" />.
+
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|analytic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769032.png" /> in an annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769033.png" /> can be represented in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769034.png" /> by a convergent Laurent series (1). In particular, in a punctured neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769035.png" /> of an isolated singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769036.png" /> of single-valued character an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769037.png" /> 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.
+
The application of Laurent series is based mainly on Laurent's theorem (1843): Any single-valued [[Analytic function|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769038.png" /> of several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769039.png" /> the following proposition can be regarded as the analogue of Laurent's theorem: Any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769040.png" />, holomorphic in the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769041.png" /> of annuli <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769042.png" />, can be represented in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769043.png" /> as a convergent multiple Laurent series
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
f ( z)  = \sum _ {| k| = - \infty } ^ { {+ }  \infty }
 +
c _ {k} ( z - a ) ^ {k} ,
 +
$$
  
 
is which the summation extends over all integral multi-indices
 
is which the summation extends over all integral multi-indices
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769045.png" /></td> </tr></table>
+
$$
 +
= ( k _ {1} \dots k _ {n} ) ,\ \
 +
| k |  = k _ {1} + \dots + k _ {n} ,
 +
$$
 +
 
 +
$$
 +
( z - a )  ^ {k}  = ( z _ {1} - a _ {1} ) ^ {k _ {1} } \dots ( z _ {n} - a _ {n} ) ^ {k _ {n} } ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769046.png" /></td> </tr></table>
+
$$
 +
c _ {k}  =
 +
\frac{1}{( 2 \pi i )  ^ {n} }
 +
\int\limits _  \gamma
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769047.png" /></td> </tr></table>
+
\frac{f ( z)  d z }{( z - a )  ^ {k+} n }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769048.png" /> is the product of the circles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769050.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769051.png" /> holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769052.png" /> cannot have isolated singularities.
+
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====
 
====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>
 
<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====
 
====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769053.png" /> be any field. The term Laurent series is also often used to denote a formal expansion of the form
+
Let $  k $
 +
be any field. The term Laurent series is also often used to denote a formal expansion of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769054.png" /></td> </tr></table>
+
$$
 +
\sum _ {i= - N } ^  \infty  a _ {i} X  ^ {i} ,\ \
 +
a _ {i} \in k ,\  N \in \mathbf Z .
 +
$$
  
 
The expressions are added termwise and multiplied as follows:
 
The expressions are added termwise and multiplied as follows:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769055.png" /></td> </tr></table>
+
$$
 +
\left ( \sum _ {i= - N } ^  \infty  a _ {i} X  ^ {i} \right )
 +
\left ( \sum _ {i= - M } ^  \infty  b _ {i} X  ^ {i} \right )  = \
 +
\sum _ {j= - N - M } ^  \infty  c _ {j} X  ^ {j} ,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769056.png" /></td> </tr></table>
+
$$
 +
c _ {j}  = \sum _ {\begin{array}{c}
 +
k+ l = j \\
 +
k,l \in \mathbf Z
 +
\end{array}
 +
} a _ {k} b _ {l}  $$
  
(note that this sum is finite). The result is a field, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769057.png" />. It is the quotient field of the ring of formal power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769058.png" />, and is called the field of formal Laurent series. A valuation is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769059.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769060.png" />. This makes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769061.png" /> a discretely valued complete field; the ring of integers is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769062.png" />, the maximal ideal is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769063.png" /> and the residue field is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769064.png" />. (Cf. also [[Valuation|Valuation]].)
+
(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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769065.png" /> is an expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057690/l05769068.png" />.
+
A Laurent polynomial over $  k $
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is an expression $  \sum _ {i = - N }  ^ {M} a _ {i} X  ^ {i} $,
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$  - N \leq  M $,  
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$  N, M \in \mathbf Z $.
  
 
More generally one also defines (formal) Laurent series in several variables and non-commutative Laurent series, cf. [[#References|[a1]]].
 
More generally one also defines (formal) Laurent series in several variables and non-commutative Laurent series, cf. [[#References|[a1]]].

Revision as of 22:15, 5 June 2020


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
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