The Winding Number
The winding number of a curveabout a pointis the number of times the curves makes a complete anticlockwise turns about the pointWe can easily label the regions into which a curves divides the complex plane in terms of winding numbers by drawing a line from the point to the part of the complex plane completely outside the curve. The winding number is equal to the number of times the curve passes the line in a counterclockwise direction minus the number of times the curve passes the line in a clockwise direction.
The winding number about a point is also equal to the total increase in argument as the curveis traversed divided byand can be found using Cauchy's Integral Formula. For a closed simple curve (with no crossings),
The integral will increase by one for each counterclockwise circuit aroundso is equal to the winding number hencefor an arbitrary closed curve