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. anti
clockwise) - 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