5.2 Winding Numbers
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It was crucial in our first version of Cauchy's theorem that the interior of the rectangle be entirely contained within . This is because, in the proof, we have no control over the locations of the subrectangles and the point they are focusing in on but must be able to apply linearization at . We would like a more general version of Cauchy's theorem - we don't want to restrict ourselves to rectangles - but must then have some way of deciding whether the inside of our contour is entirely contained within our domain . Deciding is not always simple!
It is difficult to say at first glance which of the red points in the above image are inside the contour shown and which (if any) are not. However, after some thought it may become apparent that the path can be thought of as two contours, one winding around the left-most region and one winding around the right-most region. Thus we can think of two of the red points as being outside the path.
Our goal in this section is to come up with a formal description of whether a point is inside a closed contour or not. In fact we will do slightly more and come up with a formal description for how many times a closed contour winds around a point. Let be a closed contour and let be a point that is not on . Imagine you have a piece of string. Tie one end to a pencil and place the tip of the pencil on the point . Now trace around the closed path with the other end of the piece of string. When you get back to where you started, the string will be wrapped around the pencil some number of times. This number (counted positively for anti-clockwise turns and negatively for clockwise turns) is the winding number of around and is denoted .
In examples, it is often easy to calculate winding numbers by eye and this is how we shall always do it. However, in order to use winding numbers to develop the theory of integration, we shall need an analytic expression for the winding number of a closed path around a point . Let us first consider the case .
Fix a smooth path . Lets look at on . As increases from to the argument of will vary. Sometimes it will jumo from to and sometimes it will jump from to . These jumps correspont to windings of anti-clockwise and clockwise around respectively.
Although counting jump discontinuities from the graph of is straightforward, we will need to take a slightly different perspective to develop some theory.
Definition (Continuous Argument)
Fix a path . A function is a continuous argument for if is continuous and holds for all .
The function will not usually be a continuous argument of . But if is a continuous argument of then is always an argument of .
If is a continuous argument for the closed path then
holds.
We will need the following theorem, which we will not prove.
Theorem
Every path can be written in the form
where is differentiable and is smooth and a continuous argument of .
We will finish this section with an analytic formula for the winding number.
Theorem
Fix a smooth path that is closed. Then
Proof:
By the previous theorem we can write
with differentiable and differentiable. Then
and
which is what we wanted to prove.
With the above theorem in hand, it is straightforward to prove via a substitution that
for any smooth path and any not on .
Recall that a contour is a family of smooth paths with each starting where ends. For a contour the winding number is defined to be
for any not on .