Winding number of cycle around point: Difference between revisions

Revision as of 14:42, 20 April 2008

Definition

Definition in terms of complex integrals

Let $c$ be a cycle (a sum of loops) in $C$ and $z_{0} \in C$ 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:

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

Definition in terms of homology classes

Let $c$ be a cycle (a sum of loops) in $C$ , and $z_{0} \in C$ 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 $C ∖ {z_{0}}$ .

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...)		Revision as of 14:42, 20 April 2008 (view source) Vipul (talk \| contribs) (→‎Definition in terms of homology classes) Newer edit →
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	===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>.