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The upper superexponential
#1
As it is well-known we have for
the regular superexponential at the lower fixed point.

This can be obtained by computing the Schroeder function at the fixed point of .

More precisely we set

This is a function with fixed point at 0, it is the function shifted that its fixed point is at 0.

We compute the Schroeder function of , i.e. the solution of:
where .
This has a unique analytic solution with .

Then we get the super exponential by

is adjusted such that

i.e. .

This procedure can be applied to any fixed point of .
The normal regular superexponential is obtained by applying it to the lower fixed point.

Now the upper regular superexponential is the one obtained at the upper fixed point of .
For this function we have however always ,
so the condition can not be met.
Instead we normalize it by , which gives the formula:


The interesting difference to the normal regular superexponential is that upper on is entire, while the normal one has a singularity at -2 and is no more real for .

It is entire because the inverse Schroeder function is entire, it can be continued from an initial small disk of radius r around 0 By the equation

We know that thatswhy we cover the whole complex plane with , from the initial disc around 0, and we know that is entire.

Here are some pictures of that are computed via the regular schroeder function as powerseries for our beloved base , :

   

and here the upper super exponential base 2 alone:
   
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#2
wow, how bizarre...
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#3
Ansus Wrote:But this does not satisfy the functional equation of tetration, yes?

It satisfies all except .
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#4
Ansus Wrote:So it is iterated exponential rather than tetration? Does it have asymptote?

Yes for
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#5
bo198214 Wrote:....
Instead we normalize it by , which gives the formula:


The interesting difference to the normal regular superexponential is that upper on is entire, while the normal one has a singularity at -2 and is no more real for .
....
Does this upper super expoonential equation also hold for b=?
Is this "chi" the same as the "Chi distribution" used in probability? Any links to a definition for
and
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#6
sheldonison Wrote:Does this upper super expoonential equation also hold for b=?

Interesting question. Unfortunately the convergence gets quite bad for approaching , so I could not really check numerically.
On the other hand Walker describes also two solutions for in "On the solutions of an Abelian equation". I did not really read this article, but I think he also showed that these solutions are not the limit of approaching .

Quote:Is this "chi" the same as the "Chi distribution" used in probability?
No, not at all. Its just somewhat similar to "Sch" in Schroeder.

Quote: Any links to a definition for
and

Ya, for example in the thread regular slog.
Literature is: Szekeres "Regular iteration of real and complex functions."
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#7
bo198214 Wrote:....
Unfortunately the convergence gets quite bad for approaching , so I could not really check numerically.
On the other hand Walker describes also two solutions for in "On the solutions of an Abelian equation". I did not really read this article, but I think he also showed that these solutions are not the limit of approaching .
....
in the thread regular slog.
Literature is: Szekeres "Regular iteration of real and complex functions."
Kouznetsov has graphs of the lower super exponential for in the citizendium wiki. He says "the function approaches its limiting value e, almost everywhere". I haven't seen any graphs for the upper superexponential though.

For , the function exponentially decays to its limiting value in the complex plane at +/- i . This is probably also true for the upper super exponential for , as the value at the real axis increases ...
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#8
bo198214 Wrote:As it is well-known we have for
the regular superexponential at the lower fixed point.

This can be obtained by computing the Schroeder function at the fixed point of .

More precisely we set

This is a function with fixed point at 0, it is the function shifted that its fixed point is at 0.

We compute the Schroeder function of , i.e. the solution of:
where .
This has a unique analytic solution with .

Then we get the super exponential by

is adjusted such that

i.e. .

This procedure can be applied to any fixed point of .
The normal regular superexponential is obtained by applying it to the lower fixed point.

Now the upper regular superexponential is the one obtained at the upper fixed point of .
For this function we have however always ,
so the condition can not be met.
Instead we normalize it by , which gives the formula:


The interesting difference to the normal regular superexponential is that upper on is entire, while the normal one has a singularity at -2 and is no more real for .

It is entire because the inverse Schroeder function is entire, it can be continued from an initial small disk of radius r around 0 By the equation

We know that thatswhy we cover the whole complex plane with , from the initial disc around 0, and we know that is entire.

Here are some pictures of that are computed via the regular schroeder function as powerseries for our beloved base , :

[attachment=467]

and here the upper super exponential base 2 alone:
[attachment=468]

nice post.

thanks.


regards

tommy1729
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#9
sheldonison Wrote:Kouznetsov has graphs of the lower super exponential for in the citizendium wiki. He says "the function approaches its limiting value e, almost everywhere". I haven't seen any graphs for the upper superexponential though.

I guess that the upper exponential for converges pointwise to the constant function (which of course also a solution of ).
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#10
bo198214 Wrote:As it is well-known we have for
the regular superexponential at the lower fixed point.

This can be obtained by computing the Schroeder function at the fixed point of .
.....
Now the upper regular superexponential is the one obtained at the upper fixed point of .
For this function we have however always ,
so the condition can not be met.
Instead we normalize it by , which gives the formula:

The "upper/lower" properties of these two sexp solutions are very interesting, especially being able to convert one to the other. The "upper" solution approaches the larger fixed point at -infinity, and the lower solution approaches the smaller fixed point at +infinity.

Can this be applied to Kneser's fixed point solution for bases larger than (e^(1/e))? For base e, Kneser's solution, has complex values at the real number line, and the function approaches the fixed point as x grows towards +infinity. But the desired solution has real values for all x>-2, and complex values for all x<-2 (except for the singularities). Moreover, the desired solution approaches the fixed point, as real x approaches -infinity.

This has probably already been done, but can Kneser's base e solution, approaching a complex fixed point at +infinity, be converted it to another solution, approaching the fixed point at -infinity, with real values at the real number line, for all x>-2? Perhaps this line of reasoning isn't applicable because the resulting solution, approaching the fixed point at -infinity, probably would not have imaginary values of zero for for real all x>-2.
- Sheldon
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