Contour Winding Number


The winding number of a contour gamma about a point z_0, denoted n(gamma,z_0), is defined by


and gives the number of times gamma curve passes (counterclockwise) around a point. Counterclockwise winding is assigned a positive winding number, while clockwise winding is assigned a negative winding number. The winding number is also called the index, and denoted Ind_gamma(z_0).

The contour winding number was part of the inspiration for the idea of the Brouwer degree between two compact, oriented manifolds of the same dimension. In the language of the degree of a map, if gamma:[0,1]->C is a closed curve (i.e., gamma(0)=gamma(1)), then it can be considered as a function from S^1 to C. In that context, the winding number of gamma around a point p in C is given by the degree of the map


from the circle to the circle.

See also

Complex Residue

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Krantz, S. G. "The Index or Winding Number of a Curve about a Point." §4.4.4 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 49-50, 1999.

Referenced on Wolfram|Alpha

Contour Winding Number

Cite this as:

Weisstein, Eric W. "Contour Winding Number." From MathWorld--A Wolfram Web Resource.

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