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.