Winding number of cycle around point: Difference between revisions

Latest revision as of 19:19, 18 May 2008

Definition

Definition in terms of complex integrals

Let $c$ be a cycle (a sum of loops) in $\mathbb {C}$ and $z_{0}\in \mathbb {C}$ be a point not lying on any of the loops comprising $c$ . Then, the winding number of $c$ about $z_{0}$ , denoted $n(c;z_{0})$ , is defined by:

$n(c;z_{0}):={\frac {1}{2\pi i}}\int {\frac {dz}{z-z_{0}}}$

Definition in terms of homology classes

Let $c$ be a cycle (a sum of loops) in $\mathbb {C}$ , and $z_{0}\in \mathbb {C}$ be a point not lying on any of the loos comprising $c$ . The winding number of $c$ about $z_{0}$ is the homology class of $c$ viewed as a cycle in $\mathbb {C} \setminus \{z_{0}\}$ .

Revision as of 14:41, 20 April 2008 (view source) Vipul (talk \| contribs) (New page: ==Definition== ===Definition in terms of complex integrals=== Let <math>c</math> be a cycle (a sum of loops) in <math>\mathbb{C}</math> and <math>z_0 \in \mathbb{C}</math> be a point not...)		Latest revision as of 19:19, 18 May 2008 (view source) Vipul (talk \| contribs) m (2 revisions)
(One intermediate revision by the same user not shown)
Line 9:		Line 9:
	===Definition in terms of homology classes===		===Definition in terms of homology classes===

	Let <math>c</math> be a cycle (a sum of loops) in <math>\mathbb{C}</math>, and <math>z_0 \in \mathbb{C}</math> be a point not lying on any of the loos comprising <math>c</math>. The '''winding number''' of <math>c</math> about <math>z_0</math> is the homology class of <math>c</~~mat~~> viewed as a cycle in <math>\mathbb{C} \setminus \{ z_0\}</math>.		Let <math>c</math> be a cycle (a sum of loops) in <math>\mathbb{C}</math>, and <math>z_0 \in \mathbb{C}</math> be a point not lying on any of the loos comprising <math>c</math>. The '''winding number''' of <math>c</math> about <math>z_0</math> is the homology class of <math>c</math> viewed as a cycle in <math>\mathbb{C} \setminus \{ z_0\}</math>.