Zeros and Winding Numbers

To demonstrate the zero counting of the winding number, I generated the following plot.

zeros

The plot shows the function

\(f(z) = z \, (z-\dfrac{1}{2}) \, (z+\dfrac{1}{2}) = z^3 – \dfrac{z}{4}\)

on the set

\(\{ x+iy : -1 \le x,y \le 1 \}\)

in a very specific way. The blue values represent the argument (the angle to the x-axis) of f(z), and the brightness the modulus (|f(z)|).  This looks nice, but it also has a specific meaning in the theory of complex functions.

The number of zeros of f inside a path is equal to the winding number of the image of the path under f. E.g., the unit circle is mapped to the following path with f.

zeros-2

This winds around 0 three times, exactly the number of zeros of f inside the unit circle! This fact falls from the zero counting integral for complex functions

\(\dfrac{1}{2\pi i} \oint\limits_\gamma \dfrac{f'(z)}{f(z)} \,dz = \dfrac{1}{2\pi i} \oint\limits_{f \circ \gamma} \dfrac{1}{w} \,dw.\)

If you look at the image above you can clearly see the zeros of f. Any path that goes around all three zeros once will pass the bright blue regions three times. If the path goes around two of the zeros only it winds around zero only twice. For one zero, only once.

I did that with Euler Math Toolbox (EMT) and the following code.

>rad=1; x=-rad:0.01:rad; y=x'; z=x+I*y; w=z^3-z/4;
>r=scalematrix(abs(w)); a=(sin(arg(w))+1)/2;
>C=rgb(r,r,r*(1-a)+a);
>aspect(1); fullwindow(); clg; plotrgb(C):
>aspect(2); plot3d(x,y,r,>hue,>spectral,t=linspace(0,2pi,1000); z=exp(I*t);
>aspect(1); plot2d(z^3-z/4,r=1.5):

There is also a 3D plot of |f(z)| which is not really telling much.

zeros-1

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