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I made a complex plot of the third super-root. I can't seem to get it to work for complex number with negative real part, but I think it's working for positive real part, so I've attached the plot below. The branch points are where:

superroot_3(0.731531897477381 + 0.293308661285157*I) == (0.657319327367223 + 0.704370182866530*I)

this point corresponds to where $\frac{d}{dx}x^{x^x} = 0$, and so it's not a logarithmic singularity, it's not a singularity at all, but it looks like it creates a new branch depending on which way you travel around the branch point. These branch points are shown as black dots in the plot.

[attachment=1218]
I figured out how to extend the range of the function to complex numbers with negative real part. There seems to be a third branch point at zero. The tradition of putting branch cuts towards negative infinity implies that there are really tiny strips along the negative real axis which require some shifting to a different calculation method.

[attachment=1222]
I think the glitch (at about 0.2*I) in the plot above is because I was evaluating a power series outside of its radius of convergence, I tried plotting again with a smaller estimate for the radius of convergence, and the glitch disappeared.

[attachment=1224]
(01/10/2016, 05:41 PM)andydude Wrote: [ -> ]I think the glitch (at about 0.2*I) in the plot above is because I was evaluating a power series outside of its radius of convergence, I tried plotting again with a smaller estimate for the radius of convergence, and the glitch disappeared.

How fast does the superroot3 grow on the negative real axis? Something funny happens somewhere near -35.83, maybe because the imaginary part goes to zero. Is this another singularity, or an exponential/logarithmic branch problem?
z = -0.368593375973251 + 8.24287825516783 E-7*I
z^z^z = -35.83

z = -0.36859401116538
z^z^z = -35.830698526398

(01/10/2016, 06:46 PM)sheldonison Wrote: [ -> ]How fast does the superroot3 grow on the negative real axis? Something funny happens somewhere near -35.83, maybe because the imaginary part goes to zero. Is this another singularity, or an exponential/logarithmic branch problem?
z = -0.368593375973251 + 8.24287825516783 E-7*I
z^z^z = -35.83

z = -0.36859401116538
z^z^z = -35.830698526398

I think I know what is going on here. Firstly, it appears that if we use the above branch cuts, then the imag(superroot_3(z)) would cross 0 around z = -36.

[attachment=1226]

[attachment=1227]

Secondly, the reason why this creates a discontinuity is that this corresponds to a branch cut of superpower_3 (also known as z^z^z):

[attachment=1225]

You can kind of think of traveling along the negative real axis in superroot_3 is like traveling the cyan (light blue) region in superpower_3. The dot in the plot above is approximately where the value of superpower_3 == -64, and the point would be the value of superroot_3(-64) if it was continuous, but you won't get this value on the main branch of superpower_3, because by traveling along the negative real axis of superroot_3, you've crossed a branch cut of superpower_3. In order to make that region continuous, then you would have to choose a branch of superpower_3 that is continuous with the upper-left quadrant for the "green region" above, and choose a branch of superpower_3 that is continuous with lower-left quadrant for the "blue region" above. So in order to calculate the roots of z^z^z, we need a way to choose branches of it...
It took me a while to remember the log() + 2*pi*k*I thing, after which I made this plot

[attachment=1228]

you can see how the cyan line travels from the origin into the upper-left quadrant, then into the lower-left quadrant, which can be calulated with:

$x^{x^{x^{2 \pi i/\log(x) + 1}(2 \pi i/\log(x) + 1)}(2 \pi i/\log(x) + 1)}$
Also, if you'd like to use my code, you can find the SageMath notebook on GitHub. I recently discovered that GitHub will render "ipynb" (IPython/Jupyter) notebooks as read-only, though you can download it and open it in SageMath with (sage --notebook=jupyter). I'm still learning SageMath, so I've put all my functions into one Python file that I hope to break into separate files eventually.
So I figured out how to do a (nearly) Kouznetsov style complex contour plot of the 3rd super-root (on page 12 of the attachment), and figured out how to convert a SageMath/Jupyter/IPython notebook into a pdf! I'm really starting to like Sage!
(01/17/2016, 08:26 PM)andydude Wrote: [ -> ]So I figured out how to do a (nearly) Kouznetsov style complex contour plot of the 3rd super-root (on page 12 of the attachment), and figured out how to convert a SageMath/Jupyter/IPython notebook into a pdf! I'm really starting to like Sage!

Very nice. When I have lots of time I'll need to switch over from pari-gp to Sage, and borrow your contour plot script. Being able to create a PDF would also be very nice.
(01/18/2016, 02:40 AM)sheldonison Wrote: [ -> ]Very nice. When I have lots of time I'll need to switch over from pari-gp to Sage, and borrow your contour plot script. Being able to create a PDF would also be very nice.

From what I hear, you can use pari/gp from Sage. I tried
Code:
pari("\r kneser")
from sage, to try and use your code, but it didn't work, probably because I was in the wrong directory, but I couldn't figure it out. Maybe you can get it to work, but I couldn't.
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