While working through the coursera module Introduction to Complex Analysis I wanted to develop a robust MATLAB class that allows me to visualize and gain intuition for complex valued functions.
The comp.plotter class is the result of this ambition.
Version 1.2 introduces the functionality to plot julia sets for functions of the form f(z) = z^2 + c. With this release, I learned how to work with images in MATLAB and how to export graphics to gifs.
We call the static function "julia" to display the julia set.
c = i;
comp.plotter.julia(c) % Plot the julia set of f(z) = z^2 + i
| C | Julia Set |
|---|---|
| -0.4 + 0.6i |  |
| 0.285 + 0.01i |  |
| -0.835 - 0.2321i |  |
| -0.8 + 0.156i |  |
Specific values of C were taken from the Julia Set wikipedia. I love the contrast of red and black so I plotted the julia sets using the 'hot' colormap.
What I'm most proud of is the following gif. This can be created using the juliaToGIF static function.
numSteps = 100 % number of frames in the .gif
duration = 5 % Duration in seconds
comp.plotter.juliaToGIF(numSteps, duration);
This gif was inspired by this fantastic visualization.
To instantiate a plotter class, we simply pass a function handle to the constructor. The function handle can be defined or anonymous, there is no difference.
my_fun = @(z) z^4 + 10;
p = comp.plotter(my_fun) % Equivalent to p = comp.plotter(@(z) z^4 + 10)
Upon object creation, a series of plots will automatically be generated. Let's examine them.
In order to visualize complex valued functions, we put the real part of z on the x-axis and the imag part on the y-axis. The first plot then maps the real part of f(z) to the z-axis. The second plot maps the imaginary part of f(z) to the z-axis. The third plot maps the modulus of f(z) to the z-axis. The fourth plot is the most unique - it is a domain coloring plot where the arg of f(z) is mapped to hsv values. We can use the domain coloring plot to quickly - visually - locate the zeros of our function.
The default domain of f(z) is a 100 x 100 meshed grid whose real and imaginary bounds are from -10 to 10. Let's use the zoom function to get a closer look at the function's behavior near the zeros. We can see from the domain coloring plot that the zeros are within +-2 +- 2
p.zoom(2) % The zoom function sets the new min and max bounds to the argument passed
We can see from the third plot where the modulus collapses to zero. These are the roots. While I have yet to implement a numerically precise way to find the zeros, we can get a rough estimate by selecting points in the domain coloring plot.
We can change the function by setting the f attribute of the plotter class. Doing so will automatically recalculate the output of the function. If we would like to see the same set of 4 plots, we call the plot3 function.
p.f = @(z) z^i;
p.plot3;
Funky!
Another type of plot that the comp.plotter class offers is a plot of intersecting surfaces, called using the surf function. This plot is really just a combination of the first two plots from plot3 merged into a single axis. We can see how the imaginary and real parts of the output dance together.
p.f = 1/z;
p.surf;
The comp.plotter has two static functions to build intuiton for how "fundamental" (according to me, anyways) functions appear in the complex world.
comp.plotter.trig; % Plot complex valued trig functions
comp.plotter.poly; % Plot complex valued polynomials
The compl.plotter class also has a limit function that implements the formal definition of a limit. Take for example the following trivial problem:
Let's check the comp.plotter function:
That's all I have for now, but I'd like to add more functionality as I progress through the course.
Some ideas for the future:
- zeros - a function that calculates the zeros of f(z) within epsilon
- julia - I haven't learned how to find the julia set yet, but maybe it is possible to implement it in MATLAB
- mandelbrot - A classic. Draw the mandelbrot set