(11/17/2009, 09:17 PM)bo198214 Wrote: Apart from that the pictures look good. What type of plot is it (conformal or contour?) and how is it encoded in colors? How did you compute the 4-roots?

Hue[h, s, b] is Mathematica notation for Hue-Saturation-Brightness encoding of colors, I found this function here and used it in my own code:

Code:

`PrettyHue[Indeterminate] `

:= Hue[1, 0, 0];

PrettyHue[z_] :=

Hue[N[Mod[Arg[z], 2Pi]/(2Pi)],

1/(1 + 0.3 Log[Abs[z] + 1]),

1 - 1/(1.1 + 5Log[Abs[z] + 1])];

The idea is that white = infinity, black = 0, and red = positive real, cyan/blue = negative real, and all other colors represent the angle of the complex number. Roughly speaking, if , then determines the hue, and determines the brightness. I think it is a brilliant way to show complex functions. Much more "smooth" than a contour plot, in my opinion.

Sorry I should have said more about each function. So back to the functions.

TetraRoot00 is , sorry for the silly name.

TetraRoot2 is , which requires a good CAS.

TetraPow2 is , which is pretty easy to compute.

TetraPow3 is , which is also easy to compute, but slow.

TetraPow4 is , which was very slow (attached below).

I am currently in the process of trying to work out the complex structure of TetraRoot3, or such that . Its not as "simple" as TetraRoot2, because instead of branch cuts on the real axis, the branch cuts are away from the real axis, I believe you can see this where the zeros are in TetraPow3' and if the derivative is zero, then the inverse function (TetraRoot3) should have a singularity at that point, right?

This is TetraPow3':

This is TetraPow4: