winding number

Definition: Given a closed contour γ in the complex plane (which does not meet zero), its winding number about zero is defined to be (1/2πi) intergral-around-γ 1/z dz.

This simply counts the number of times that γ winds around zero in the positive sense (ie. anticlockwise) - this can be seen from the fact that an antiderivative of 1/z is log(z), which jumps in value by 2πi every time it goes around zero in this sense.
This notion is of use in The Principle of the Argument, and the winding number of a contour is precisely the number if corresponds to in The Fundamental Group of C\{0}.

The winding number of a closed curve C in the plane around a point is intuitively the number of times the curve goes around the point. It is possible to give a formula for winding numbers in terms of line integrals of complex functions. This turns out to be important in complex analysis, since allows you to interpret some integrals geomtrically, as in e.g. the residue theorem. Let us first give a naive definition:

Definition 1: Let C : [0, 1] → C be a closed curve that does not pass through the origin (C is the complex plane. Working here rather than the real plane gives neater notation, and anyway one of the main points is to get to the complex integral formula). Let A be a continuous choice of argument for C, i.e. a continuous function A : [0, 1] → R such that A(t) is a choice of argument for C(t) for all t. The winding number of C about the origin is n(C) = (A(1)-A(0))/2π. (The winding number around any other point is defined analogously.)

Strictly speaking we need to check that such continuous choices of argument exist and that the difference A(1)-A(0) is independent of the choice of A. Basically, this follows from the fact that the map R → S¹ is a covering map and the path-lifting property of coverings.

There is an alternative, equivalent, definition in terms of a contour integral.

Definition 2: The winding number around the origin of a closed curve C in C (that does not pass through the origin) is

n(C) = (2πi)^-1∫_C z^-1 dz

To prove that the definitions are equivalent, let L(t) = L(0) + ∫^t C(t)^-1C'(t)dt, where L(0) is a choice of logarithm for C(0). (d/dt)(C(t)*exp(-L(t)) = (C'(t) - C(t)*C(t)^-1*C'(t))*exp(-L(t)) = 0, so exp(L(t)) = C(t) for all t. Thus L is a continuous choice of logarithm of C, and im L is a continuous choice of argument for C. Hence

n(C) = (2π)^-1im(L(1)-L(0)) = (2πi)^-1∫_C z^-1 dz

as desired.

If the curve C is changed continuously without crossing the origin then the winding number is constant (it varies continuously in the integers, which is a discrete set). Thus the winding number is well-defined on homotopy classes of curves in C\{0}. In algebraic topology terms, the winding number defines a map from the fundamental group of C\{0} to the integers. It is easy to see that this map is a homomorphism, and also that it is bijective; in other words it is an isomorphism.