Winding number
From Encyclopedia of Mathematics
Let 
be an arc in the complex plane and let 
be a point not on 
. A continuous argument of 
on 
is a continuous real-valued function 
on 
that for each 
is an argument of 
, i.e. 
for some 
. Such functions can be found, and if 
, 
are two continuous arguments, then they differ by a constant integral multiple of 
. It follows that the increase of the argument, 
, does not depend on the choice of the continuous argument. It is denoted by 
. If 
is a piecewise-regular arc,
 |
In the special case that 
is a closed curve, i.e. 
, 
is necessarily an integral multiple of 
and the integer
 |
is called the winding number of 
with respect to 
. For a piecewise-regular closed curve 
with 
not on 
one has
 |
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
Winding number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Winding_number&oldid=49226