# Tetration Forum

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(08/02/2009, 10:47 AM)bo198214 Wrote: [ -> ]However our widely used super-logarithm does not fit in this pattern,
thatswhy I followed Dmitriis suggestion: arcsuper-exponential

This is the inverse of the super-exponential and hence what we previously called super-logarithm. arcsuper-exponential means also Abel function of the exponential, so we *dont* introduce an extra arcsuperfunction.

I think "arcsuper" is a terrible idea. I personally dislike the "arctan" terminology, and prefer "inverse tangent" instead. What's wrong with "super-logarithm"? (aside from being a terrible Google search term)

(08/02/2009, 10:47 AM)bo198214 Wrote: [ -> ]@Andrew: Instead of with matrices I would like you to start with the equations, because the matrices are not really necessary to explain the approach. Something roughly like: From the Abel equation ... powerseries ... infinite equation system ... (represented with the Carleman matrix as ...) ambiguity of solutions .... intuitive solution ... etc.

Hmm... I have no idea how I would describe it without matrices... if you can do that, more power to you. For me it would be like trying to describe functions without sets. However, it would be good to generalize a little, because the definition of the Abel matrix doesn't apply to doing intuitive iteration of (x -> a x) because you have to use different truncations for this case. I remember we tried to use the Abel matrix for this (x -> a x) a while ago, and for the longest time, I've been wanting to follow through with this. It seems like even knowing the solution (log_a(x)) doesn't really make it any easier when discussing this example.

Andrew Robbins
Quote:But super logarithm is not a superfunction of the logarithm, but an inverse superfunction of the exponential.

The superfunction of the logarithm is F(z)= tet(-z) if F(0)=1, so i don't think it will matter, but...

edit: just call it superlogarithm, that's what everyone calls it anyway
(08/04/2009, 04:50 AM)andydude Wrote: [ -> ]The "super" terminology has always meant: a rank-4 function that is analogous to a rank-3 function.

Or a rank-n+1 function that is analogous to a rank-n function.
Well but for this usage we dont need the super-terminology.
As long as we are inside the operation ladder that starts with addition (or increment) we can always say tetra-exponential, tetra-logarithm (as Tetratophile already suggested), tetra-root. No need for any "super".

The meaning of "super" in this configuration is also quite cumbersome, to determine the meaning of "super-bla" one has to do the following steps:
1. determine the rank of bla
2. determine the type of bla (power,exponential, logarithm)
3. increment the rank, keep the type.

So outside the hierarchy "super" would not make any sense (no rank, no type).
On the other hand a terminology for the inverse Abel function is desperately needed and it has to be applicable to functions outside the hierarchy.
"Inverse Abel function of the exponential" is just not usable compared to "super-exponential".
And "super" in this new context can be defined very precisely for arbitrary functions.
And it matches the expectation, the super-exponential grows much faster than the exponential (while the old "tetra-logarithm" grows much slower than the logarithm, so one would call it rather sub-logarithm. Like Hoohsmand named his operation ultra exponential and infra logarithm, if I remember correctly).

So summarizing "super" is not needed in the default operation ladder, but a term for "inverse Abel function" is desperately needed outside the operation ladder and "super" is intuitively clear for that usage.

Quote: I have a feeling that the push for consistent terminology will leave the corpus of writings on this forum in a state of complete inconsistency.

Yes your are right. And I promise you I will never again change the name of the "intuitive iteration", but if a terminology has deficiencies thats just the course of history that it will be replaced especially if it just starts to develop.

Quote: I vote for "superlogarithm" or "Abel function of exponential". No "arcsuper".

superlogarithm can be replaced by tetra-logarithm.
"Abel function of exponential" is correct but too long!
Then make a counter suggestion for "Abel function of foo"!
(08/04/2009, 04:46 PM)bo198214 Wrote: [ -> ]superlogarithm can be replaced by tetra-logarithm.
"Abel function of exponential" is correct but too long!
Then make a counter suggestion for "Abel function of foo"!

The "Tetra-abelian" ... ?

Hmmm
(08/04/2009, 04:46 PM)bo198214 Wrote: [ -> ][quote='andydude' pid='3534' dateline='1249357846']
....Yes your are right. And I promise you I will never again change the name of the "intuitive iteration", but if a terminology has deficiencies thats just the course of history that it will be replaced especially if it just starts to develop.

Quote: I vote for "superlogarithm" or "Abel function of exponential". No "arcsuper".

superlogarithm can be replaced by tetra-logarithm.
"Abel function of exponential" is correct but too long!
Then make a counter suggestion for "Abel function of foo"!
Super-logarithm and slog should be recognized synonyms of the inverse super-exponential function (or tetra-logarithm). The only other thing super-logarithm could possibly refer to would be the super-exponential function, mirrored along the imaginary axis so that the $+\infty$ and the $-\infty$ are swapped, and that isn't a terribly interesting notation.
(08/04/2009, 09:17 PM)sheldonison Wrote: [ -> ]...Super-logarithm and slog should be recognized synonyms of the inverse super-exponential function (or tetra-logarithm). The only other thing super-logarithm could possibly refer to would be the super-exponential function, mirrored along the imaginary axis so that the $+\infty$ and the $-\infty$ are swapped, and that isn't a terribly interesting notation.
Super-logarithm and slog are very confusive terms. I suggest to use "arctetration" and "ate" instead. Better to keep term "Super-logarithm" for the superfunction of logarithm. I have not yet plotted such a function, but it is doable. Also, one can make SuperCos, SuperTetrational, and so on. While, we have the ArcFactorial and SuperFactorial http://www.ils.uec.ac.jp/~dima/PAPERS/2009supefae.pdf
If anybody wants to suggest some better terminology, now is good moment.

P.S. If you have some Ficial, we can make a SuperFicial for it.
But do not tell economists about our achievements: they deal with Inflation...
(08/05/2009, 08:16 AM)Kouznetsov Wrote: [ -> ]Super-logarithm and slog are very confusive terms. I suggest to use "arctetration" and "ate" instead.
Hear, hear! And may I suggest, while we're at it, that we do away with the term "logarithm" altogether. How are we supposed to remember that it's the inverse of exponentiation? Better to call it arcexponentiation, for consistency. Indeed, division and subtraction should likewise be arcmultiplication and arcaddition. Let's get all these confusing terms out of the mathematical glossary and be done with it!

On a sidenote, I've never really minded switching freely between the terms "tetration" and "super-exponentiation", yet for some reason the term "super-logarithm" (or simply "s-log" or "slog") seems stuck in my vocabulary, and the next closest term that comes to mind is "the inverse of the super-exponential function" or something similarly descriptive, but lacking brevity.

So "superlogarithm" it shall remain. You will have to pry it from my cold dead hands!

Or try to come up with something a little more natural (pun intended) than arc-super-exponentiation or arc-tetration. Sorry, as a native english speaker and amateur mathemetician (because the professionals don't count ), the "arc"s just aren't cutting it. But maybe it will grow on me over time... I suppose part of the problem is that I've never really preferred the "arc-" prefixes for the trigonometric functions (much less for non-trigonometric functions!), even though they have their benefits when, e.g., the $\sin^{-1}(x)$ notation is ambiguous (multiplicative or functional inverse?).

Then again, what does the "arc" really mean? In the context of trigonometry, which was founded on the study of actual "arcs" among other things, I never really put much thought into it. Oddly, wikipedia and several online dictionaries fail to make any etymological note whatsoever on the use of "arc".
Jay, I ask you to be more constructive.
The problems are explained, if you have solutions we are keen to hear.
But, I dont want to have polemics on the board.

Yes, indeed I think the "arc" prefix for trigonometric functions comes from measuring the length of the arc on the unit circle (which is the angle).
So etymologically this would count against a usage as "inverse"-prefix.
(08/05/2009, 06:15 PM)bo198214 Wrote: [ -> ]Jay, I ask you to be more constructive.
The problems are explained, if you have solutions we are keen to hear.
But, I dont want to have polemics on the board.

Yes, indeed I think the "arc" prefix for trigonometric functions comes from measuring the length of the arc on the unit circle (which is the angle).
Sorry, perhaps the sarcasm didn't come through, or perhaps was viewed as aggressive, when it was meant in good jest. Apologies if I was offensive.
It was not yet offensive, I just want to avoid that it develops in that direction.

So if you keep "super-logarithm" what prefix do you use for inverse Abel function?
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