Namespaces
Variants
Actions

Difference between revisions of "Winding number"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w0980201.png" /> be an arc in the complex plane and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w0980202.png" /> be a point not on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w0980203.png" />. A continuous argument of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w0980204.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w0980205.png" /> is a continuous real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w0980206.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w0980207.png" /> that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w0980208.png" /> is an [[Argument|argument]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w0980209.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w09802010.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w09802011.png" />. Such functions can be found, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w09802012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w09802013.png" /> are two continuous arguments, then they differ by a constant integral multiple of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w09802014.png" />. It follows that the increase of the argument, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w09802015.png" />, does not depend on the choice of the continuous argument. It is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w09802016.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w09802017.png" /> is a piecewise-regular arc,
+
<!--
 +
w0980201.png
 +
$#A+1 = 29 n = 0
 +
$#C+1 = 29 : ~/encyclopedia/old_files/data/W098/W.0908020 Winding number
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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/w/w098/w098020/w09802018.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
In the special case that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w09802019.png" /> is a closed curve, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w09802020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w09802021.png" /> is necessarily an integral multiple of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w09802022.png" /> and the integer
+
Let  $  \Gamma = \{ {z( \tau ) } : {\alpha \leq  \tau \leq  \beta } \} $
 +
be an arc in the complex plane and let  $  c $
 +
be a point not on  $  \Gamma $.  
 +
A continuous argument of  $  z- c $
 +
on  $  \Gamma $
 +
is a continuous real-valued function  $  \phi $
 +
on  $  [ \alpha , \beta ] $
 +
that for each  $  \tau \in [ \alpha , \beta ] $
 +
is an [[Argument|argument]] of  $  z ( \tau ) - c $,  
 +
i.e. $  z ( \tau ) - c = r  \mathop{\rm exp} ( i \phi ( \tau )) $
 +
for some  $  r $.  
 +
Such functions can be found, and if  $  \phi ( \tau ) $,
 +
$  \psi ( \tau ) $
 +
are two continuous arguments, then they differ by a constant integral multiple of $  2 \pi $.  
 +
It follows that the increase of the argument,  $  \phi ( \beta ) - \phi ( \alpha ) $,
 +
does not depend on the choice of the continuous argument. It is denoted by  $  [  \mathop{\rm arg}  z ( \tau ) - c ] _  \Gamma  $.  
 +
If  $  \Gamma $
 +
is a piecewise-regular arc,
  
<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/w/w098/w098020/w09802023.png" /></td> </tr></table>
+
$$
 +
[  \mathop{\rm arg}  z ( \tau ) - c ] _  \Gamma  = \
 +
\mathop{\rm Im}  \int\limits _  \Gamma 
 +
\frac{1}{z-}
 +
c  dz .
 +
$$
  
is called the winding number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w09802024.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w09802025.png" />. For a piecewise-regular closed curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w09802026.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w09802027.png" /> not on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098020/w09802028.png" /> one has
+
In the special case that  $  \Gamma $
 +
is a closed curve, i.e. $  z ( \alpha ) = z ( \beta ) $,
 +
$  [  \mathop{\rm arg}  z ( \tau ) - c ] _  \Gamma  $
 +
is necessarily an integral multiple of  $  2 \pi $
 +
and the integer
  
<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/w/w098/w098020/w09802029.png" /></td> </tr></table>
+
$$
 +
n ( \Gamma , c )  =
 +
\frac{1}{2 \pi }
 +
[  \mathop{\rm arg}  z( \tau ) - c] _  \Gamma  $$
 +
 
 +
is called the winding number of  $  \Gamma $
 +
with respect to  $  c $.
 +
For a piecewise-regular closed curve  $  \Gamma $
 +
with  $  c $
 +
not on  $  \Gamma $
 +
one has
 +
 
 +
$$
 +
n ( \Gamma , c )  =
 +
\frac{1}{2 \pi i }
 +
\int\limits _  \Gamma 
 +
\frac{1}{z-}
 +
c  dz .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Henrici,  "Applied and computational complex analysis" , '''1''' , Wiley (Interscience)  (1974)  pp. §4.6</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Henrici,  "Applied and computational complex analysis" , '''1''' , Wiley (Interscience)  (1974)  pp. §4.6</TD></TR></table>

Revision as of 08:29, 6 June 2020


Let $ \Gamma = \{ {z( \tau ) } : {\alpha \leq \tau \leq \beta } \} $ be an arc in the complex plane and let $ c $ be a point not on $ \Gamma $. A continuous argument of $ z- c $ on $ \Gamma $ is a continuous real-valued function $ \phi $ on $ [ \alpha , \beta ] $ that for each $ \tau \in [ \alpha , \beta ] $ is an argument of $ z ( \tau ) - c $, i.e. $ z ( \tau ) - c = r \mathop{\rm exp} ( i \phi ( \tau )) $ for some $ r $. Such functions can be found, and if $ \phi ( \tau ) $, $ \psi ( \tau ) $ are two continuous arguments, then they differ by a constant integral multiple of $ 2 \pi $. It follows that the increase of the argument, $ \phi ( \beta ) - \phi ( \alpha ) $, does not depend on the choice of the continuous argument. It is denoted by $ [ \mathop{\rm arg} z ( \tau ) - c ] _ \Gamma $. If $ \Gamma $ is a piecewise-regular arc,

$$ [ \mathop{\rm arg} z ( \tau ) - c ] _ \Gamma = \ \mathop{\rm Im} \int\limits _ \Gamma \frac{1}{z-} c dz . $$

In the special case that $ \Gamma $ is a closed curve, i.e. $ z ( \alpha ) = z ( \beta ) $, $ [ \mathop{\rm arg} z ( \tau ) - c ] _ \Gamma $ is necessarily an integral multiple of $ 2 \pi $ and the integer

$$ n ( \Gamma , c ) = \frac{1}{2 \pi } [ \mathop{\rm arg} z( \tau ) - c] _ \Gamma $$

is called the winding number of $ \Gamma $ with respect to $ c $. For a piecewise-regular closed curve $ \Gamma $ with $ c $ not on $ \Gamma $ one has

$$ n ( \Gamma , c ) = \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{1}{z-} c dz . $$

References

[a1] P. Henrici, "Applied and computational complex analysis" , 1 , Wiley (Interscience) (1974) pp. §4.6
How to Cite This Entry:
Winding number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Winding_number&oldid=49226