# Tetration Forum

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I thought I'd introduce myself, and explain how I got here.

I found this forum through Wikipedia, my interest in tetration dates back a long time, although I didn't know the name.

Sometime when I was a graduate student in mathematics at CalTech, Richard Feynman used to wander into the math lounge and ask weird questions. Most of them either had obvious answers or obviously (to me, anyway), had no answers. However, one question that he asked was whether there is a "natural" increasing functions f and g such that
(a) f(f(x)) = e^x,
(b) g(x+1)=e^g(x)

(Again, it was obvious even then to me, that if there is a solution of (b) for g, then
: f(x) = g(1/2+g^(-1)(x))
is a solution of (a) , and that there are many C^(infinity) solutions of (b)with g(0)=1, and g defined on (-2, +infinity). (I might have selected g(0)=0, but it doesn't really matter, right.)

In the notation of this forum, g(x) = e^^x. As I pointed out over on Wikipedia, if there is a real-analytic solution of (b), there are many, as if h is real-analytic, period 1, and "small" (to satisfy "increasing"), then g(x+h(x)) also satisfies condition (b).

And I don't really know much more than that. I hope that, together with some of the other posters, we can find some sort of "natural" tetration.
(07/01/2009, 02:27 AM)Arthur Rubin Wrote: [ -> ]And I don't really know much more than that. I hope that, together with some of the other posters, we can find some sort of "natural" tetration.

Hi Arthur -
that would be nice...

Saw your contributions in wikipedia: welcome! Have a good time here

Gottfried
(07/01/2009, 08:21 AM)Gottfried Wrote: [ -> ]
(07/01/2009, 02:27 AM)Arthur Rubin Wrote: [ -> ]And I don't really know much more than that. I hope that, together with some of the other posters, we can find some sort of "natural" tetration.

Hi Arthur -
that would be nice...

Saw your contributions in wikipedia: welcome! Have a good time here

Gottfried

yeah welcome Arthur.

what wikipedia contribution ? link ?
(07/01/2009, 02:27 AM)Arthur Rubin Wrote: [ -> ]I thought I'd introduce myself, and explain how I got here.

Welcome Athur!
There are a lot methods already developed for real analytic tetration.
However not much is proved, especially not whether these methods are equal in the end.
Have a look through the forum to get a glimpse.
Wikipedia huh......... I think we may have met before.........
Indeed, welcome to the forum.

I hope you find your tetration. Everybody seems to find their own at some point. The hard part is discovering how each are related. I think the holy grail here is to prove that two independently-defined tetrations are equivalent.

Also, in this forum, "g" is called a superexponential (or tetrational) function and "g^-1" is called a superlogarithm. I'm sure if you search for these terms on this forum, you will find many many discussions.

Andrew Robbins
wait a minute ??

are you the same dude who blocks me on the wiki page of tetration ?

http://en.wikipedia.org/wiki/Talk:Tetration

where you can see Arthur Rubin pretending to be an expert on tetration and blocking my method based upon " using sinh ".

Hell , they even recommended him this forum , and mentioned my method is on this site , which he * mysteriously * ignored !!

Turns out is reading here , and even posting here , before the tetration debate even started !

and suddenly he is not an expert anymore and wants to be friend !?

what kind of a fool do you think i am ??

im really pissed now.
so Arthur Rubin

you have not posted anything about tetration in all this time !!

you claim to be an expert on tetration , even before that word existed.

i guess you dont want to provide evidence for that huh ?

you are extremely silent about tetration and extremely silent in this forum ?

are you sick ? or just not an expert as you claim ?

im not making this up :

from wikipedia tetration talk page :

" And, for what it's worth, I worked on the functional square root of the exponential function long before the term tetration was coined. — Arthur Rubin (talk) 02:39, 11 January 2011 "

also i wonder what you mean by :

" Furthermore, even if the resulting function exists, it almost certainly fails regularity conditions, and would produce different functions if one replaced 2 sinh(x) by 2 e^-a sinh(x+a), for any real a. "

( ignoring the handwaving " almost certainly " etc instead of math )

it seems you missed the point that 'a' needs to be 0 in order to have the usefull UNIQUE fixpoint at 0 to compute the iterations of 2 sinh(x) and have a strictly increasing entire taylor series for it.

to be specific you need [(e^a)^2 - 1] / (e^a)^2 (*) to be 0 in order to have f(0) = 0 ( the fixpoint of 2 sinh(x) ) hence a = 0.

( for the clever people , note that a = +oo - the other solution(*) - already reduces to f(0) = 1 and exp(x) ! )

and another thing , my method is not levy's.

you need to do some math to rewrite it as a special case of levy , but in essence it is not.

you might as well have said it relates to abel equation and that abel equation is nothing new.

furthermore it were the people on this forum who first mentioned levy , not Rubin.

( and those people did get published , whereas Rubin cannot show that concerning tetration )

guess you will be quit again.

i dont need to be quit , unlike you , because i have nothing to hide.

no claimed published papers i can not prove.

no people saying they thought of it first.

and no lies or double identities or hidden agendas.

tommy1729