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
 

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