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}.