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''analytic arc''
 
''analytic arc''
  
A curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a0122101.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a0122102.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a0122103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a0122104.png" />, which has an analytic parametrization. This means that the coordinates of its points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a0122105.png" /> can be expressed as analytic functions of a real parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a0122106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a0122107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a0122108.png" />, i.e. in a certain neighbourhood of each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a0122109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221010.png" />, the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221011.png" /> can be represented as convergent power series in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221012.png" />, and the derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221014.png" />, do not simultaneously vanish at any point of the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221015.png" />. This last condition is sometimes treated separately, and an analytic curve which satisfies it is called a regular analytic curve. An analytic curve is called closed if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221017.png" />.
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A curve $  K $
 +
in an $  n $-
 +
dimensional Euclidean space $  \mathbf R  ^ {n} $,  
 +
$  n \geq  2 $,  
 +
which has an analytic parametrization. This means that the coordinates of its points $  x = ( x _ {1} \dots x _ {n} ) $
 +
can be expressed as analytic functions of a real parameter $  x _ {i} = x _ {i} (t) $,  
 +
$  i = 1 \dots n $,  
 +
$  \alpha \leq  t \leq  \beta $,  
 +
i.e. in a certain neighbourhood of each point $  t _ {0} $,  
 +
$  \alpha \leq  t _ {0} \leq  \beta $,  
 +
the functions $  x _ {i} (t) $
 +
can be represented as convergent power series in $  t - t _ {0} $,  
 +
and the derivatives $  x _ {i}  ^  \prime  (t _ {0} ) $,  
 +
$  i = 1 \dots n $,  
 +
do not simultaneously vanish at any point of the segment $  [ \alpha , \beta ] $.  
 +
This last condition is sometimes treated separately, and an analytic curve which satisfies it is called a regular analytic curve. An analytic curve is called closed if $  x _ {i} ( \alpha ) = x _ {i} ( \beta ) $,  
 +
$  i = 1 \dots n $.
  
An analytic curve in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221018.png" /> of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221019.png" /> can be represented as a complex-analytic function of a real parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221022.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221023.png" />. If the analytic curve is located in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221024.png" />, then a conformal mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221025.png" /> into any domain will also yield an analytic curve. If the set of intersection points of two analytic curves is infinite, these analytic curves coincide.
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An analytic curve in the plane $  \mathbf C = \mathbf C  ^ {1} $
 +
of the complex variable $  z = x _ {1} + ix _ {2} $
 +
can be represented as a complex-analytic function of a real parameter $  z = f(t) $,  
 +
$  \alpha \leq  t \leq  \beta $,  
 +
$  f ^ {\ \prime } (t) \neq 0 $
 +
on $  [ \alpha , \beta ] $.  
 +
If the analytic curve is located in a domain $  D \subset  \mathbf C $,  
 +
then a conformal mapping of $  D $
 +
into any domain will also yield an analytic curve. If the set of intersection points of two analytic curves is infinite, these analytic curves coincide.
  
In general, in a complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221027.png" />, the complex coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221028.png" /> of the points of an analytic curve can be represented as analytic functions of a real parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221031.png" />. It should be noted, however, that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221032.png" />, the term  "analytic curve"  may sometimes denote an [[Analytic surface|analytic surface]] of complex dimension one.
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In general, in a complex space $  \mathbf C  ^ {n} $,  
 +
$  n \geq  1 $,  
 +
the complex coordinates $  z _ {i} $
 +
of the points of an analytic curve can be represented as analytic functions of a real parameter $  z _ {i} = z _ {i} (t) $,  
 +
$  \alpha \leq  t \leq  \beta $,  
 +
$  i = 1 \dots n $.  
 +
It should be noted, however, that if $  n > 1 $,  
 +
the term  "analytic curve"  may sometimes denote an [[Analytic surface|analytic surface]] of complex dimension one.
  
On a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221033.png" /> an analytic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221034.png" /> can be represented as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221035.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221036.png" /> is a local uniformizing parameter of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221037.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221039.png" /> is an analytic function of a real parameter in a neighbourhood of any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012210/a01221040.png" />.
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On a Riemann surface $  S $
 +
an analytic curve $  K $
 +
can be represented as $  f(t) = \psi ( \phi (t)) $,  
 +
where $  z = \psi (P) $
 +
is a local uniformizing parameter of the points $  P $
 +
on $  S $
 +
and $  f(t) $
 +
is an analytic function of a real parameter in a neighbourhood of any point $  t _ {0} \in [ \alpha , \beta ] $.
  
 
====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)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1969)  pp. Chapt. 3  (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" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1969)  pp. Chapt. 3  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.C. Gunning,  H. Rossi,  "Analytic functions of several complex variables" , Prentice-Hall  (1965)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.C. Gunning,  H. Rossi,  "Analytic functions of several complex variables" , Prentice-Hall  (1965)</TD></TR></table>

Latest revision as of 18:31, 5 April 2020


analytic arc

A curve $ K $ in an $ n $- dimensional Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, which has an analytic parametrization. This means that the coordinates of its points $ x = ( x _ {1} \dots x _ {n} ) $ can be expressed as analytic functions of a real parameter $ x _ {i} = x _ {i} (t) $, $ i = 1 \dots n $, $ \alpha \leq t \leq \beta $, i.e. in a certain neighbourhood of each point $ t _ {0} $, $ \alpha \leq t _ {0} \leq \beta $, the functions $ x _ {i} (t) $ can be represented as convergent power series in $ t - t _ {0} $, and the derivatives $ x _ {i} ^ \prime (t _ {0} ) $, $ i = 1 \dots n $, do not simultaneously vanish at any point of the segment $ [ \alpha , \beta ] $. This last condition is sometimes treated separately, and an analytic curve which satisfies it is called a regular analytic curve. An analytic curve is called closed if $ x _ {i} ( \alpha ) = x _ {i} ( \beta ) $, $ i = 1 \dots n $.

An analytic curve in the plane $ \mathbf C = \mathbf C ^ {1} $ of the complex variable $ z = x _ {1} + ix _ {2} $ can be represented as a complex-analytic function of a real parameter $ z = f(t) $, $ \alpha \leq t \leq \beta $, $ f ^ {\ \prime } (t) \neq 0 $ on $ [ \alpha , \beta ] $. If the analytic curve is located in a domain $ D \subset \mathbf C $, then a conformal mapping of $ D $ into any domain will also yield an analytic curve. If the set of intersection points of two analytic curves is infinite, these analytic curves coincide.

In general, in a complex space $ \mathbf C ^ {n} $, $ n \geq 1 $, the complex coordinates $ z _ {i} $ of the points of an analytic curve can be represented as analytic functions of a real parameter $ z _ {i} = z _ {i} (t) $, $ \alpha \leq t \leq \beta $, $ i = 1 \dots n $. It should be noted, however, that if $ n > 1 $, the term "analytic curve" may sometimes denote an analytic surface of complex dimension one.

On a Riemann surface $ S $ an analytic curve $ K $ can be represented as $ f(t) = \psi ( \phi (t)) $, where $ z = \psi (P) $ is a local uniformizing parameter of the points $ P $ on $ S $ and $ f(t) $ is an analytic function of a real parameter in a neighbourhood of any point $ t _ {0} \in [ \alpha , \beta ] $.

References

[1] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)
[2] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1969) pp. Chapt. 3 (In Russian)

Comments

References

[a1] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965)
How to Cite This Entry:
Analytic curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_curve&oldid=16087
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article