Encyclopedia > Winding number

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Winding number

Intuitively, the winding number of a curve &gamma with respect to a point z0 is the number of times &gamma goes around z0 in a counter-clockwise direction.

Formally, it is defined as follows:

If &gamma is a closed curve in C and z0 is a point in C not on &gamma, then the winding number of &gamma with respect to z0 (alternately called the index of &gamma with respect to z0) is defined by the formula

I(&gamma, z0) = 1/(2&pii) ∫γ 1/(z-z0) dz.

The winding number is used primarily in complex analysis in the Residue theorem.

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