# Tetration Forum

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While I was making complex plots, I usually exported to PDF, then converted PDF to PNG, which made lots of white lines everywhere in the image.I realized that the pictures looked bad when zoomed out, so I rewrote the complex plot function to write directly to a PNG file. The images look much better.

See the zip files below.
(11/17/2009, 07:51 PM)andydude Wrote: [ -> ]While I was making complex plots, I usually exported to PDF, then converted PDF to PNG, which made lots of white lines everywhere in the image.I realized that the pictures looked bad when zoomed out, so I rewrote the complex plot function to write directly to a PNG file. The images look much better.

See the zip files below.

But Andrew, its make no sense to convert it to png if you dont show it inline. If you make an attachment you can do it as pdf. If you want to inline you need a png. Somehow you now chose the worst scenario: a png as attachment inside a zip.

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?
(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 $z = r e^{i \theta}$, then $\theta$ determines the hue, and $r$ 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 $f(z) = z^{1/z}$, sorry for the silly name.
TetraRoot2 is $f(z) = \ln(z)/W(\ln(z))$, which requires a good CAS.

TetraPow2 is $f(z) = z^z$, which is pretty easy to compute.
TetraPow3 is $f(z) = z^{z^z}$, which is also easy to compute, but slow.
TetraPow4 is $f(z) = z^{z^{z^z}}$, which was very slow (attached below).

I am currently in the process of trying to work out the complex structure of TetraRoot3, or $f(z)$ such that ${}^{3}(f(z)) = z$. 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':
[attachment=649]

This is TetraPow4:
[attachment=650]
(11/17/2009, 09:17 PM)bo198214 Wrote: [ -> ]Somehow you now chose the worst scenario: a png as attachment inside a zip.

The reason for that is your file size restrictions. Some of the images are >1MB, so it will not let me upload. Apparently there is a 500KB limit on PNGs and 1MB limit on zips...
Some more plots. I will try and inline this time...

TetraLogE is $f(z) = \text{slog}_e(z)$, I have adopted Kouznetsov's branch structure, since it corresponds to the direction that log "naturally" goes (pun intended):
[attachment=651]

TetraLogE' is $f'(z)$, which I did some Photoshoping on, sorry.
[attachment=652]

TetraExpE is $f(z) = \exp_e^z(1)$, using Kouznetsov's expansion at 3i:
[attachment=653]

TetraExpE' is $f'(z)$:
[attachment=654]

TetraExpE-z is $f(z) = \exp_e^z(1) - z$, which shows the fixed-points of tetration as black dots:
[attachment=655]