Winding number
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 |
Winding number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Winding_number&oldid=49226