### A dynamic example of the argument principle

The Argument Principle deals with the behavior of simple closed curves
under "mappings" by functions f(z) which are analytic and *not
zero* on the closed curve, but which may have zeros and poles
inside the simple closed curve. The tool for investigating this is the
integral over the curve of the function f´(z)/f(z), the "logarithmic
derivative" of f(z).
Suppose C is such a closed curve, and f(z) is such a function. Inside
C, the function f´(z)/f(z) has isolated singularities at each
pole and each zero. The type of those isolated singularities of
f´(z)/f(z) at each of those points turns out to be just a simple
pole, that is, a pole of order 1. The residue at each point is **+(the order of the zero)** and **-(the order of the pole)**. The Argument Principle
applies the Residue Theorem to the integral over C of f´(z)/f(z)
and then further analyzes the integral over C of that function. It
declares that the integral is the amount that the argument of f(z)
would change (increase or decrease) as z moves along C. The reason
this is true is that many different logs (as antiderivatives of
f´(z)/f(z), the integrand) can be used to "evaluate" the integral
along C.

**An example**
Suppose f(z) is the function :

** [(z+1)**^{3}(z-i)^{2}]
----------------
[(z+3)^{3}(z-1)^{1}].

This function has zeros of order 3 at -1 and order 2 at i. It has
poles of order 3 at -3 and 1 at 1. So
- The Laurent series of f´(z)/f(z) centered at -1 in a disc
centered at -1 begins with the term 3/(z+3)+higher order terms (powers
of z+3).
- The Laurent series of f´(z)/f(z) centered at i in a disc
centered at i begins with the term 2/(z-i)+higher order terms (powers
of z-i).
- The Laurent series of f´(z)/f(z) centered at -3 in a disc
centered at -3 begins with the term 3/(z+3)+higher order terms (powers
of z+2).
- The Laurent series of f´(z)/f(z) centered at 1 in a disc
centered at 1 begins with the term 1/(z-1)+higher order terms.

A sort of picture of these points is shown to the right. Now I want to
investigate what happens to the following simple closed curve:
s(t)=2e^{iπt} with t in the interval [0,2Pi]. This is, of
course, a circle with radius 2 and center 0. So the circle encloses
three of the four points that are shown. The count *should* be
-1+2+3=4, and this, multiplied by 2πi, should be the increase in
the argument of f(z) around C. The image curve in the complex plane is
shown in an animation created using `Maple`. If you are not using a fast connection,
then I apologize. The file size is half a megabyte! The picture on the
right is a still picture of the image of the circle. That may be
easier for people to see and study. I know as I try to count and see
how much the curve winds around 0, I almost got dizzy!

Do you **believe** the theorem? I will admit
that getting a nice-looking example took some experimentation. A great
deal of fussing was needed to see the loops. With a really "random"
example, the loops are all different scales, some large, some small.

**
Maintained by **`
greenfie@math.rutgers.edu` and last modified 4/22/2008.