## Complex Function Viewer

This tool visualizes any complex-valued
function as a conformal map by assigning a color to each point in
the complex plane according to the function's value at that point.

Enter any expression in z.

The identity function z shows how colors are assigned:
a gray ring at |z| = 1 and a black and
white circle around any zero
and colored circles around 1,
i, -1,
and -i.
Checkers cover the plane in a 1/16th
unit grid.
Colors are turquoise in the
positive direction,
red in the negative, gold-green
towards +i, purplish
towards -i,
and darker towards infinity.
There is also a colored circle towards infinity at
|z| > 16
that can be seen at any pole towards infinity such as
in 1/z.

Here are some example functions to try:

z^2

zz*

(z+1)/(z-1)

sin(z)

e^z

log(z)

sech(z)

arctan(z)

z^3-1

0.926(z+7.3857e-2 z^5+4.5458e-3 z^9)

Jacobi elliptic sn(z, 0.3)

Gamma function gamma(z)

Iterated function iter(z+z'^2,z,12)

## Animating Conformal Maps

To visualize the relationships within families of complex functions,
parameterize them with the variables t, u, s, r, or n. The tool will render
a range of complex functions for values of the parameter, adjustable with
a slider or shown in an aimation. The parameter t will vary linearly
from 0 to 1; u will circle through complex units; s follows a sine wave
between -1 and 1; r follows a sine wave from 0 to 1 and back; and
n counts integers from 1 to 60.

For example, to see the relationship between z^3 and
z^3+1, simply view:

z^3+t

On the globe, multiplying by powers of unity will rotate the world on its
axis:

u(z-i)/(z+i)

Because more than 300 frames are computed, parameterized expressions can
take a long time to fully render. A rough, blurry sketch is drawn quickly,
and finer-grained rendering will follow for several minutes. When done,
the frames will be antialiased and animated at 24 fps.

Simple families of rational function produce mesmerizing animations:

z^2+s

z^3+1+u

z^5+uz+1

z^2/(r+z)

Iterated functions and sums can also be animated. For example, the
following are well-known Taylor series for
e^z,
sin(z),
1/(1-z), and
log(1-z):

sum(z^n/n!)

sum((-1)^n/(2n+1)! z^(2n+1))

sum(z^n)

sum(z^(n+1)/(n+1))

The radii of convergence can clearly be seen
in the last two examples

tool by
David Bau
Modified by Derin Sherman