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Principal Branch of the Super-logarithm
#1
I have been playing around with the complex equivalent of a piecewise-defined slog, and with the sections of it that should appear on its 0th branch. I feel like what Jay calls the "backbone" along with power series along the real axis, should be in the 0th branch. Since the real part of the real axis is generally greater than the backbone, the analytic continuation from the positive real axis towards the backbone (going through a line with imaginary part approximately pi) should bring us to the 1st branch. An analytic continuation from the backbone (going through a line with imaginary part approximately pi) should then bring us to the -1st branch. Where the branch cuts are is really arbitrary, and is anyone's guess as to where would be best. This is what I'm writing this post about, because I don't know which branch cut is nicest.

In the images below, the big dots in the the branch-cut diagram are the fixed points of e^x. The smaller dots are the 2 pi shifting of the fixed points, which are always a singularity on some branch. The thick lines are the branch cuts.

Branch cut system A (branch cuts, real part, imaginary part)
seems to be the best of all of these, but does not preserve periodicity. It has the fewest branch cuts, and does not avoid the second pair of fixed points which the others do avoid. All consider the first pair of fixed points, and the singularities derived from them. If the second pair of fixed points do become a problem (I haven't tried extending the domain that far yet) then we can just make branch cuts from that pair, away from the real line).

Branch cut system B (branch cuts, real part, imaginary part)
has the nicest real-part graph, and preserves periodicity across the imaginary axis. All branch systems should have a periodic backbone, but where the periodicity ends is a matter of taste, I suppose.

Branch cut system C (branch cuts, real part, imaginary part)
was actually my first attempt, because this is how Jay describes the rotation around the first pair of fixed points. The thing I like about this one is that it contains both real functions. The real function along the real axis, and the real function along the (real + ipi) axis, which has real output since:



I'm sorry I haven't taken the time to convert these images to png, but knowing you guys, you'd probably ask for pdf anyways...

Andrew Robbins
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#2
Since you brought up the non-primary fixed points, it's worth mentioning that they aren't truly "fixed points" on the slog I've been working with. The easiest way to describe them is as fixed endpoints of loops. For example, connect a line from the second fixed point, 2.0623+7.5886*I, to its image 2.0623+1.3054*I. Now expontiate every point on the line, and you'll get a circle with endpoints at 2.0623+7.5886*I. Exponentiate again, and you'll get a complex (literally and descriptively) looping curve with endpoints at 2.0623+7.5886*I. At integer intervals, you arrive back at the same point, but continuous iteration in between is not well defined.

Going back to the regular slog as Henyrk described it, it's relatively easy to derive for bases between 1 and eta. The upper and lower fixed points give different solutions, but interestingly enough, the solution at the lower fixed point looks a heck of a lot like the logarithmic branches of bases greater than eta. The solution at the upper fixed point looks a heck of a lot like the exponential branches of bases greater than eta. I think the reason that bases greater than eta can have both types is mainly due to the complex fixed points. The regular slog at one complex fixed point or another would, for example, never go through the origin, because you can't reach it with logarithms (all fixed points are attracting when performing logarithms), and you can't reach it with exponentials (because 0 is not in the output of the exponential). But when defined, mysterious as it still is to me, as an slog using both primary fixed points, we are able to reach the origin, and indeed, this is how we know that the slog isn't the same as the regular slog as Henryk defined it, because it can be solved at the origin.

For example, the regular slog for base sqrt(2), defined for the upper fixed point at z=4, is undefined at the origin. Starting with small concentric circles centered at z=4, and exponentiating them, we eventually start to wrap around the origin. But as contorted as these wrappings may get, they never reach the origin. This is the same behavior as can be seen on the exponential branches of the slog base e, which I haven't done any pictures for yet. I suppose I should get some made up.

As far as branch cuts, I'd have to think about them more. I've been viewing it sort of as a collection of screws, with singularities as the axes of the screws, and the threads connecting to each other as one continuous sheet. I then let the branch cuts move with an "observer" embedded in the sheet, such that the branch cuts lie behind the singularities as seen by this observer. I let this "observer" wander through the slog, with branch cuts shifting as necessary so that he never sees the branch cuts, just the singularities and the two different worlds to the "left" and "right" of those singularities. I haven't given much thought to well-defined and somewhat "static" branch cuts...
~ Jay Daniel Fox
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#3
So that means are only fixed points of the exponential function, and not fixed-points of the principal branch of the logarithm, since it takes it takes the modulus (modulo 2ipi) of everything so it surrounds the real axis. The Imaginary part of which is well beyond the range of the principal branch of the logarithm. Since I am using the principal branch of the logarithm to move points with positive real part into the unit circle, these fixed points would not show up on this branch as singularities. If there was another branch that was using the branch of the logarithm in which these points are fixed points, then of course we would need another branch cut for them, since they would appear as singularities. But as it is, I think branch cut system A is the simplest choice for the principal branch of the base-e super-logarithm.

Andrew Robbins
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#4
I have taken the time to convert the images to png, so here they are.

For branch system A:
[Image: slog_A_real.png] [Image: slog_A_imag.png]

For branch system B:
[Image: slog_B_real.png] [Image: slog_B_imag.png]

For branch system C:
[Image: slog_C_real.png] [Image: slog_C_imag.png]

Any suggestions on which to use? Or is it too early to be deciding on branch cuts?

Andrew Robbins
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#5
Well, it's not as simple as saying, "Here are some cuts". There are a few rules you'll have to follow:

1. Branch cuts must be between pairs of singularities, or between a singularity and infinity, or from infinity to infinity. This is the simple rule we all know and love.
2. The branch cuts cannot pass through an impassable boundary. At most, they can go through an endpoint. Branch cut systems B and C violate this rule.

Thus, only system A would even qualify as a valid system. In that sense, it works fine.

My preference is to draw branch cuts between every "proper" pair of singularities. A "proper" pair of singularities will be logarithmicized/exponentiated images of the line between the primary pair of singularities, or the line between the upper primary singularity and the 2*pi*I offset of the lower primary singularity. This will get all the branch cuts necessary when going into successively "deeper" logarithmic branches. There will be distinct sheets that will be surround on all sides by such branch cuts, or by infinity. The "backbone" is the most basic, with branch cuts to the right, along the imaginary line with real part 0.31813, and infinity to the left, top, and bottom. Within the first layer of logarithmic branches, you'll have a branch cut on the left on an imaginary line segment with real part 0.31813, then curves along the top and bottom that connect at a point at infinity. The next layer in gets more complicated, but in essense the principle is the same.

However, exponential branches will require branch cuts similar to your System A. They get more complicated once you've wrapped around a singularity, because of the weird singularities at the integer tetrates of the base (e.g., 0, 1, e, e^e, etc.), or their logarithmicized/exponentiated images, as the case may be.

I'll draw pictures of what I mean when I get a chance.
~ Jay Daniel Fox
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#6
Do you mean like this:
   
If so, then the series expansion about the origin has nowhere to go. If you mean this without the big red line, then that would be similar to branch cut system C, which you said was invalid, or was this because it wasn't lined up with the singularities?

Andrew Robbins
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#7
jaydfox Wrote:(from this thread)
Exponentiate any point on the left boundary, and we'll get a point on the right boundary. Conversely, take the logarithm of any point on the right boundary, and we'll end up with a point on the left boundary. And of course, the fixed points complete the enclosure of this region.

Using this knowledge, that "big red line" is the logarithm of the straight line between the two primary fixed points. And if we consider the last one Branch System D, then we can still use Branch System A as the standard branch cuts. Or in other words, if is Branch System A, then would be Branch System D!

Andrew Robbins
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#8
(11/17/2007, 10:58 AM)andydude Wrote:

i do not think that is correct ...

it is tempting to substitute x => x + pi and similar tricks but in general



is false. take for example x-> x^3



is false because slog is not a "modified" inverse super of !

so i think this is a mistake ... i would prefer a better explaination then my own very much Dodgy

perhaps this is the reason that we do not have many ( or many known ? ) calculus identities related to tetration ; substitution is troublesome , hence together with the growth speeds , integrals are harder or even impossible to express in closed form.

i feel im missing something here and im still a bit confused about this...

( although i can unfortunately not ask for a concrete question - apart from the one below - , just an elaboration )

i must say i wonder about the existance of the following

h and g do not commute :



derivate



substitute



integrate



( * notice how h'(x) is on both sides as if it does not matter if we differentiate with respect to it , or just substitute directly , which mainly motives this question btw )

???
***

it appears that " -1 " is a function of f and g in any , or at least my imagination tells me.

so maybe we should define , or could define for some functions :



or



or something similar ??

so another question becomes





regards

tommy1729
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