Winding number

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

Formally, the winding number is defined as follows:

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

I(γ, z₀) = 1/(2πi) ∫_γ 1/(z – z₀) dz

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

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Winding number