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riemann surface for tetration and slog?
#1
Assuming that tetration and slog are holomorphic for all applicable z, what doe the reimann surface for tetration and superlogarithm look like?
superlogarithm has cultines, and there are several options to plotting them. klouznetsov's graphs at citizendium shows that the cutlines can be rotated at arbitrary angles. What happens if you rotate the left-hand graph's cutlines by pi/2? or the right hand graph's cutlines by -pi/2? Can someone produce a parametric surface plot for superlogarithm and tetration? (at least a quick, dirty approximation for it)
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#2
(06/24/2009, 12:45 PM)Tetratophile Wrote: Assuming that tetration and slog are holomorphic for all applicable z, what doe the reimann surface for tetration and superlogarithm look like?

The interesting thing with the slog is, that new branchpoints come into existence on different branches (which then create new branches), which you can see in Dmitriis plot.
I am not sure whether anyone has a complete description of this infinitely nested structure.

At least we can say that there must be infinitely many branchpoints of the slog, because each branch point occurs again at for all integer .
.
However they dont need to occur on the same branch, but they must occur on some branch.

And I never heard about anyone considering the branchpoint structure of sexp at .
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#3
todo:

find tet z, find inverse, find all the branchpoints...


I don't htink the cut at z<-2 of tet z is a branch cut. it is where from all directions the function blows up?

Because of the logarithm involved in superexponential it may be a branch cut after all.
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#4
(07/25/2009, 04:38 AM)Tetratophile Wrote: .. find tet z, find inverse, find all the branchpoints...
The inverse of tetration is arctetration; some of its Riemann surfaces are plotted at
http://www.ils.uec.ac.jp/~dima/PAPERS/2009fractae.pdf

As for the Riemann surfaces of tetration, they are not so spectacular. I post the complex map of modified tetratin
   
levels of constant and those of are shown with thick lines for integer values. In this function, the cut line (dashed) from the point is directed vertically, to ; the other cut still runs horisontally, left along the real axis.
In the upper halfplane, , as well as at , this function coincides with the conventional tetration, plotted previously. Note that the only imaginaty part of has jump at the vertical cut. The real part remains continuous at this cut.

Quote:I don't htink the cut at z<-2 of tet z is a branch cut. it is where from all directions the function blows up?
Perhaps, you wanted to say "branch point".
The function is not equal to its Taylor series, developed in its branchpoint.
(Even it the series exist.) The function has no need to blow up in vicinity of the branch point. For example, the remains bounded in vicinity of its branch points.

Henryk, may I post here the plots of and in order to show that a function has no need to blow up in vicinity of its branch point?
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#5
as you go left along the real axis: singularity at z=-2, and then bounce back up, then branch cut? so confusing... why?

of the tetration's riemann surface, i expected a logarithmic branch point because of the logarithm's involvement in tetration - i expected the ray z<=-2 is a cut of the superlogarithm.

Kouznetsov Wrote:some of its Riemann surfaces are plotted at
http://www.ils.uec.ac.jp/~dima/PAPERS/2009fractae.pdf
I was looking for something like this (a parametric plot of the imaginary part of a function) at wikipedia:
http://en.wikipedia.org/wiki/File:Rieman...ce_log.jpg

Kouznetsov Wrote:Henryk, may I post here the plots of sqrt(exp) and sqrt(!) in order to show that a function has no need to blow up in vicinity of its branch point?

Yes, I understand that a function doesn't need to blow up near the branch point. The square root function is an example that has a branch point at 0.

But I was expecting something like a logarithmic branch cut.
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#6
   
(07/29/2009, 08:20 AM)Tetratophile Wrote: ..
I was looking for something like this (a parametric plot of the imaginary part of a function) at wikipedia:
http://en.wikipedia.org/wiki/File:Rieman...ce_log.jpg
...
But I was expecting something like a logarithmic branch cut.
Tetratophile, in the case of many branchpoints, the structure is more complicated.
While I can show what happens if you rotate two and three cutlines:
   
You may try to plot the real or imaginary part of the function as a surface in the 3-dimentional space using some Mathematica software. Due to the countable set of intersections, I expect such a surface to look like a foam, http://upload.wikimedia.org/wikipedia/co...lastic.jpg
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