Wednesday, December 3, 2008

Fractal Results

These are the results of the requests I got for Newton's fractals. Each function generates a fractal that, in principle, covers the entire plane, but I only show a small window of it. When I talk about the "range" of the fractal, I am referring to the location and dimensions of the window I chose.

Susan asked for the hyperbolic trig functions. Actually, they look more or less the same as the regular trig functions, but that's okay because the trig functions turn out well.

This is the function cosh(z) in the range -.5 to .5 on the real (horizontal) axis, and -.5 to .5 on the imaginary (vertical) axis. Note that only the first six roots get unique colors--the rest are all black.

And this is the function tanh(z) in the range -3 to 3 in the real axis and -2 to 2 in the imaginary axis.

An anonymous commenter asked for the function e^-(ixcosx)+e^(xsinx). This is a complicated one, graphed from 0 to 2 in the real and imaginary axes. I suspect those black comb-shaped things are actually artifacts of my program, but it took such a long time to generate the fractal that I wasn't going to try to figure out how to get rid of them. Besides, they look cool.