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Pretty pictures.
#3
(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:
   


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Messages In This Thread
Pretty pictures. - by andydude - 11/17/2009, 07:51 PM
RE: Pretty pictures. - by bo198214 - 11/17/2009, 09:17 PM
RE: Pretty pictures. - by andydude - 11/18/2009, 05:56 AM
RE: Pretty pictures. - by andydude - 11/18/2009, 05:59 AM
RE: Pretty pictures. - by andydude - 11/18/2009, 06:12 AM

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