A point z₀ and a curve C

In mathematics, the winding number is a topological invariant playing a leading role in complex analysis. An image of a contour and a point to illustrate the winding number File links The following pages link to this file: Winding number Categories: BSD images ... An image of a contour and a point to illustrate the winding number File links The following pages link to this file: Winding number Categories: BSD images ... Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... In the mathematical field of topology a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. ... Complex analysis is the branch of mathematics investigating holomorphic functions, i. ...

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. In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ...

In the image on the right, the winding number of the curve (C) about the inner point pictured (z₀) is 3, since the curve makes three full revolutions around the point. The small loop on the left does not go around the point and so has no effect overall. Note that if the direction of the curve were reversed, the winding number would be −3 instead of 3.

Formal definitions

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:

This is verifiable from applying the Cauchy integral formula — the integral will be a multiple of 2πi, since each time γ goes about z₀, we have effectively calculated the integral again. Cauchys integral formula is a central statement in complex analysis. ... This article deals with the concept of an integral in calculus. ...

The winding number is used in the residue theorem. The residue theorem in complex analysis is a powerful tool to evaluate path integrals of meromorphic functions over closed curves and can often be used to compute real integrals as well. ...

In more abstract terms, the fundamental group of the complement of a point P in the plane is infinite cyclic. Choose a generator σ in the positively-oriented direction, of the fundamental group with base point some fixed point Q ≠ P. Create a loop based at Q from C, by joining Q to C by an arc to the starting point of C, going round c, then going back the same way to Q. The winding number will be m if the class of this loop in the fundamental group is mσ. In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...

See also

• Residue theorem
• Argument principle

The residue theorem in complex analysis is a powerful tool to evaluate path integrals of meromorphic functions over closed curves and can often be used to compute real integrals as well. ... The contour C (black), the zeros of f (blue) and the poles of f (red). ...

External links

Winding number (http://planetmath.org/?op=getobj&from=objects&id=3291) on PlanetMath. PlanetMath is a free, collaborative, online mathematics encyclopedia. ...