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''local uniformizer, local parameter''
 
''local uniformizer, local parameter''
  
A complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l0602401.png" /> defined as a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l0602402.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l0602403.png" /> on a [[Riemann surface|Riemann surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l0602404.png" />, defined everywhere in some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l0602405.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l0602406.png" /> and realizing a homeomorphic mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l0602407.png" /> onto the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l0602408.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l0602409.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024010.png" /> is said to be a distinguished or parametric neighbourhood, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024011.png" /> a distinguished or parametric mapping, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024012.png" /> a distinguished or parametric disc. Under a parametric mapping any point function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024013.png" />, defined in a parametric neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024014.png" />, goes into a function of the local uniformizing parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024015.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024016.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024018.png" /> are two parametric neighbourhoods such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024019.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024021.png" /> are the two corresponding local uniformizing parameters, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024022.png" /> is a univalent holomorphic function on some subdomain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024023.png" /> realizing a biholomorphic mapping of this subdomain into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024024.png" />.
+
A complex variable $  t $
 +
defined as a continuous function $  t _ {p _ {0}  } = \phi _ {p _ {0}  } ( p) $
 +
of a point $  p $
 +
on a [[Riemann surface|Riemann surface]] $  R $,  
 +
defined everywhere in some neighbourhood $  V ( p _ {0} ) $
 +
of a point $  p _ {0} \in R $
 +
and realizing a homeomorphic mapping of $  V ( p _ {0} ) $
 +
onto the disc $  D ( p _ {0} ) = \{ {t \in \mathbf C } : {| t | < r ( p _ {0} ) } \} $,  
 +
where $  \phi _ {p _ {0}  } ( p _ {0} ) = 0 $.  
 +
Here $  V ( p _ {0} ) $
 +
is said to be a distinguished or parametric neighbourhood, $  \phi _ {p _ {0}  } : V ( p _ {0} ) \rightarrow D ( p _ {0} ) $
 +
a distinguished or parametric mapping, and $  D ( p _ {0} ) $
 +
a distinguished or parametric disc. Under a parametric mapping any point function $  g ( p) $,  
 +
defined in a parametric neighbourhood $  V ( p _ {0} ) $,  
 +
goes into a function of the local uniformizing parameter $  t $,  
 +
that is, $  g ( p) = g [ \phi _ {p _ {0}  }  ^ {-} 1 ( t) ] = G ( t) $.  
 +
If $  V ( p _ {0} ) $
 +
and $  V ( p _ {1} ) $
 +
are two parametric neighbourhoods such that $  V ( p _ {0} ) \cap V ( p _ {1} ) \neq \emptyset $,  
 +
and $  t _ {p _ {0}  } $
 +
and $  t _ {p _ {1}  } $
 +
are the two corresponding local uniformizing parameters, then $  t _ {p _ {1}  } = \phi _ {p _ {1}  } [ \phi _ {p _ {0}  }  ^ {-} 1 ( t _ {p _ {0}  } )] $
 +
is a univalent holomorphic function on some subdomain of $  D ( p _ {0} ) $
 +
realizing a biholomorphic mapping of this subdomain into $  D ( p _ {1} ) $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024025.png" /> is the Riemann surface of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024027.png" /> is a regular element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024028.png" /> with projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024030.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024031.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024032.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024033.png" /> is a singular, or algebraic, element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024034.png" />, corresponding to the [[Branch point|branch point]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024035.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024036.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024037.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024039.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024040.png" />. In a parametric neighbourhood of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024041.png" /> the local uniformizing parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024042.png" /> actually realizes a local [[Uniformization|uniformization]], generally speaking, of the many-valued relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024043.png" />, according to the formulas (for example, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024044.png" />):
+
If $  R = R _ {F} $
 +
is the Riemann surface of an analytic function $  w = F ( z) $
 +
and $  p _ {0} $
 +
is a regular element of $  F ( z) $
 +
with projection $  z _ {0} \neq \infty $,  
 +
then $  t _ {p _ {0}  } = z - z _ {0} $;  
 +
$  t _ {p _ {0}  } = 1 / z $
 +
for $  z _ {0} = \infty $.  
 +
If $  p _ {0} $
 +
is a singular, or algebraic, element of $  F ( z) $,  
 +
corresponding to the [[Branch point|branch point]] $  z _ {0} $
 +
of order $  k - 1 > 0 $,  
 +
then $  t _ {p _ {0}  } = ( z - z _ {0} )  ^ {1/k} $
 +
for $  z _ {0} \neq \infty $
 +
and $  t _ {p _ {0}  } = 1 / z  ^ {1/k} $
 +
for $  z _ {0} = \infty $.  
 +
In a parametric neighbourhood of an element $  p _ {0} $
 +
the local uniformizing parameter $  t $
 +
actually realizes a local [[Uniformization|uniformization]], generally speaking, of the many-valued relation $  w = F ( z) $,  
 +
according to the formulas (for example, for $  z _ {0} \neq \infty $):
  
<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/l060/l060240/l06024045.png" /></td> </tr></table>
+
$$
 +
= z _ {0} + t  ^ {k} ,\ \
 +
= F ( z _ {0} + t  ^ {k} )  = w ( t) ,\ \
 +
k \geq  1 .
 +
$$
  
In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024046.png" /> is a Riemann surface with boundary, for points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024047.png" /> belonging to the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024048.png" /> the local uniformizing parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024049.png" /> maps the parametric neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024050.png" /> onto the half-disc
+
In the case when $  R $
 +
is a Riemann surface with boundary, for points $  p _ {0} $
 +
belonging to the boundary of $  R $
 +
the local uniformizing parameter $  t _ {p _ {0}  } = \phi _ {p _ {0}  } ( p) $
 +
maps the parametric neighbourhood $  V ( p _ {0} ) $
 +
onto the half-disc
  
<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/l060/l060240/l06024051.png" /></td> </tr></table>
+
$$
 +
D ( p _ {0} )  = \
 +
\{ {t \in \mathbf C } : {| t | < r ( p _ {0} ) ,  \mathop{\rm Im}  t \geq  0 } \}
 +
.
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024052.png" /> is a a [[Riemannian domain|Riemannian domain]] over a complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024054.png" />, then the local uniformizing parameter
+
If $  R $
 +
is a a [[Riemannian domain|Riemannian domain]] over a complex space $  \mathbf C  ^ {n} $,  
 +
$  n > 1 $,  
 +
then the local uniformizing parameter
  
<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/l060/l060240/l06024055.png" /></td> </tr></table>
+
$$
 +
t _ {p _ {0}  }  = \
 +
\phi _ {p _ {0}  } ( p)  = \
 +
( t _ {1} \dots t _ {n} ) _ {p _ {0}  }  = \
 +
( \phi _ {1} ( p) \dots \phi _ {n} ( p) ) _ {p _ {0}  }
 +
$$
  
realizes a homeomorphic mapping of the parametric neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024056.png" /> onto the polydisc
+
realizes a homeomorphic mapping of the parametric neighbourhood $  V ( p _ {0} ) $
 +
onto the polydisc
  
<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/l060/l060240/l06024057.png" /></td> </tr></table>
+
$$
 +
D ( p _ {0} ) =
 +
$$
  
<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/l060/l060240/l06024058.png" /></td> </tr></table>
+
$$
 +
= \
 +
\{ t = ( t _ {1} \dots t _ {n} ) \in \mathbf C  ^ {n} : | t _ {1} | < r _ {1} ( p _ {0} ) \dots
 +
| t _ {n} | < r _ {n} ( p _ {0} ) \} .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024059.png" /> is not empty, then the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024060.png" /> biholomorphically maps a certain subdomain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024061.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060240/l06024062.png" />.
+
If $  V ( p _ {0} ) \cap V ( p _ {1} ) $
 +
is not empty, then the mapping $  t _ {p _ {1}  } = \phi _ {p _ {1}  } [ \phi _ {p _ {0}  }  ^ {-} 1 ( t _ {p _ {0}  } ) ] $
 +
biholomorphically maps a certain subdomain of $  D ( p _ {0} ) $
 +
into $  D ( p _ {1} ) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''2''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Springer,  "Introduction to Riemann surfaces" , Addison-Wesley  (1957)  pp. Chapt.10</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''2''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Springer,  "Introduction to Riemann surfaces" , Addison-Wesley  (1957)  pp. Chapt.10</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.M. Farkas,  I. Kra,  "Riemann surfaces" , Springer  (1980)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.M. Farkas,  I. Kra,  "Riemann surfaces" , Springer  (1980)</TD></TR></table>

Revision as of 22:17, 5 June 2020


local uniformizer, local parameter

A complex variable $ t $ defined as a continuous function $ t _ {p _ {0} } = \phi _ {p _ {0} } ( p) $ of a point $ p $ on a Riemann surface $ R $, defined everywhere in some neighbourhood $ V ( p _ {0} ) $ of a point $ p _ {0} \in R $ and realizing a homeomorphic mapping of $ V ( p _ {0} ) $ onto the disc $ D ( p _ {0} ) = \{ {t \in \mathbf C } : {| t | < r ( p _ {0} ) } \} $, where $ \phi _ {p _ {0} } ( p _ {0} ) = 0 $. Here $ V ( p _ {0} ) $ is said to be a distinguished or parametric neighbourhood, $ \phi _ {p _ {0} } : V ( p _ {0} ) \rightarrow D ( p _ {0} ) $ a distinguished or parametric mapping, and $ D ( p _ {0} ) $ a distinguished or parametric disc. Under a parametric mapping any point function $ g ( p) $, defined in a parametric neighbourhood $ V ( p _ {0} ) $, goes into a function of the local uniformizing parameter $ t $, that is, $ g ( p) = g [ \phi _ {p _ {0} } ^ {-} 1 ( t) ] = G ( t) $. If $ V ( p _ {0} ) $ and $ V ( p _ {1} ) $ are two parametric neighbourhoods such that $ V ( p _ {0} ) \cap V ( p _ {1} ) \neq \emptyset $, and $ t _ {p _ {0} } $ and $ t _ {p _ {1} } $ are the two corresponding local uniformizing parameters, then $ t _ {p _ {1} } = \phi _ {p _ {1} } [ \phi _ {p _ {0} } ^ {-} 1 ( t _ {p _ {0} } )] $ is a univalent holomorphic function on some subdomain of $ D ( p _ {0} ) $ realizing a biholomorphic mapping of this subdomain into $ D ( p _ {1} ) $.

If $ R = R _ {F} $ is the Riemann surface of an analytic function $ w = F ( z) $ and $ p _ {0} $ is a regular element of $ F ( z) $ with projection $ z _ {0} \neq \infty $, then $ t _ {p _ {0} } = z - z _ {0} $; $ t _ {p _ {0} } = 1 / z $ for $ z _ {0} = \infty $. If $ p _ {0} $ is a singular, or algebraic, element of $ F ( z) $, corresponding to the branch point $ z _ {0} $ of order $ k - 1 > 0 $, then $ t _ {p _ {0} } = ( z - z _ {0} ) ^ {1/k} $ for $ z _ {0} \neq \infty $ and $ t _ {p _ {0} } = 1 / z ^ {1/k} $ for $ z _ {0} = \infty $. In a parametric neighbourhood of an element $ p _ {0} $ the local uniformizing parameter $ t $ actually realizes a local uniformization, generally speaking, of the many-valued relation $ w = F ( z) $, according to the formulas (for example, for $ z _ {0} \neq \infty $):

$$ z = z _ {0} + t ^ {k} ,\ \ w = F ( z _ {0} + t ^ {k} ) = w ( t) ,\ \ k \geq 1 . $$

In the case when $ R $ is a Riemann surface with boundary, for points $ p _ {0} $ belonging to the boundary of $ R $ the local uniformizing parameter $ t _ {p _ {0} } = \phi _ {p _ {0} } ( p) $ maps the parametric neighbourhood $ V ( p _ {0} ) $ onto the half-disc

$$ D ( p _ {0} ) = \ \{ {t \in \mathbf C } : {| t | < r ( p _ {0} ) , \mathop{\rm Im} t \geq 0 } \} . $$

If $ R $ is a a Riemannian domain over a complex space $ \mathbf C ^ {n} $, $ n > 1 $, then the local uniformizing parameter

$$ t _ {p _ {0} } = \ \phi _ {p _ {0} } ( p) = \ ( t _ {1} \dots t _ {n} ) _ {p _ {0} } = \ ( \phi _ {1} ( p) \dots \phi _ {n} ( p) ) _ {p _ {0} } $$

realizes a homeomorphic mapping of the parametric neighbourhood $ V ( p _ {0} ) $ onto the polydisc

$$ D ( p _ {0} ) = $$

$$ = \ \{ t = ( t _ {1} \dots t _ {n} ) \in \mathbf C ^ {n} : | t _ {1} | < r _ {1} ( p _ {0} ) \dots | t _ {n} | < r _ {n} ( p _ {0} ) \} . $$

If $ V ( p _ {0} ) \cap V ( p _ {1} ) $ is not empty, then the mapping $ t _ {p _ {1} } = \phi _ {p _ {1} } [ \phi _ {p _ {0} } ^ {-} 1 ( t _ {p _ {0} } ) ] $ biholomorphically maps a certain subdomain of $ D ( p _ {0} ) $ into $ D ( p _ {1} ) $.

References

[1] A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) (Translated from Russian)
[2] G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10
[3] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian)

Comments

References

[a1] H.M. Farkas, I. Kra, "Riemann surfaces" , Springer (1980)
How to Cite This Entry:
Local uniformizing parameter. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_uniformizing_parameter&oldid=11803
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article