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The 14 pages that link to Winding number, each with the reason it gives.

• Cauchy's integral formulaRelated: For general contours, the formula’s value is multiplied by the contour’s winding number.

• Fundamental groupBroader topic: For the circle, it identifies loop classes with integers.

• Jordan curve theoremRelated: It distinguishes the two complementary regions by assigning different winding numbers.

• Complex logarithmRelated: Each winding around zero changes a continued logarithm by \(2\pi i\).

• Residue theoremRelated: For contours that wind repeatedly, each residue is weighted by the winding number.

• Argument principleRelated: The image contour’s winding around zero is the argument change counted by the theorem.

• Contour integrationRelated: It weights each enclosed singularity in the residue theorem.

• ResidueRelated: For contours with repeated winding, each enclosed residue is weighted by this number.

• Cauchy's integral theoremRelated: For domains with holes, winding around excluded points helps determine whether the theorem applies.

• Closed curveRelated: It records a loop’s turning around points not on the curve.

• Poincaré–Hopf theoremBroader topic: In two dimensions, a zero’s index is the winding number of the field around it.

• Rouché's theoremRelated: The theorem preserves the boundary curve's winding around zero during deformation.

• Chern numberCompared with: It is another integer topological invariant, often defined from a lower-dimensional map.

• Star polygonRelated: It helps describe how a self-intersecting star outline loops around its center.