The upper superexponential

[bo198214]

As it is well-known we have for \( b<e^{1/e} \)
the regular superexponential at the lower fixed point.

This can be obtained by computing the Schroeder function at the fixed point \( a \) of \( F(x)=b^x \).

More precisely we set
\( G(x)=F(x+a)-a = b^{x+a}-a=b^a b^x -a = a b^x -a = a(b^x-1) \)
This is a function with fixed point at 0, it is the function \( F \) shifted that its fixed point is at 0.

We compute the Schroeder function \( \chi \) of \( G \), i.e. the solution of:
\( \chi(G(x))=c\chi(x) \) where \( c=G'(0)=a\ln(b)=\ln(b^a)=\ln(a) \).
This has a unique analytic solution with \( \chi'(0)=1 \).

Then we get the super exponential by
\( \operatorname{sexp}_b(t)=a+\chi^{-1}(c^x \chi(y_0) \)
\( y_0 \) is adjusted such that
\( 1=\operatorname{sexp}_b(0)=a+\chi^{-1}(\chi(y_0))=a+y_0 \)
i.e. \( y_0=1-a \).

This procedure can be applied to any fixed point \( a \) of \( b^x \).
The normal regular superexponential is obtained by applying it to the lower fixed point.

Now the upper regular superexponential \( \operatorname{usexp} \) is the one obtained at the upper fixed point of \( b^x \).
For this function we have however always \( \operatorname{usexp}(x)>a \),
so the condition \( \operatorname{usexp}(0)=1 \) can not be met.
Instead we normalize it by \( \operatorname{usexp}(0)=a+1 \), which gives the formula:
\( \operatorname{usexp}_b(t)=a+\chi^{-1}\left(\ln(a)^x \chi(1)\right) \)

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 \( x<-2 \).

It is entire because the inverse Schroeder function \( \chi^{-1} \) is entire, it can be continued from an initial small disk of radius r around 0 By the equation
\( \chi^{-1}(c^n x)=G^{[n]}(\chi(x)) \)
We know that \( c>1 \) thatswhy we cover the whole complex plane with \( c^nx \), \( x \) from the initial disc around 0, and we know that \( G^{[n}] \) is entire.

Here are some pictures of \( \operatorname{sexp} \) that are computed via the regular schroeder function as powerseries for our beloved base \( b=\sqrt{2} \), \( a=2,4 \):

   

and here the upper super exponential base 2 alone:
   

[andydude]

wow, how bizarre...

[bo198214]

But this does not satisfy the functional equation of tetration, yes?

It satisfies all except \( f(0)=1 \).

[bo198214]

So it is iterated exponential rather than tetration? Does it have asymptote?

Yes \( y=4 \) for \( x\to -\infty \)

[sheldonison]

....
Instead we normalize it by \( \operatorname{usexp}(0)=a+1 \), which gives the formula:
\( \operatorname{usexp}_b(t)=a+\chi^{-1}\left(\ln(a)^x \chi(1)\right) \)

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 \( x<-2 \).
....

Does this upper super expoonential equation also hold for b=\( e^{1/e} \)?
Is this "chi" the same as the "Chi distribution" used in probability? Any links to a definition for
\( \chi \) and \( \chi^{-1} \)

[bo198214]

Does this upper super expoonential equation also hold for b=\( e^{1/e} \)?

Interesting question. Unfortunately the convergence gets quite bad for \( b \) approaching \( e^{1/e} \), so I could not really check numerically.
On the other hand Walker describes also two solutions for \( b=e^{1/e} \) 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 \( e^{1/e} \).

Is this "chi" the same as the "Chi distribution" used in probability?

No, not at all. Its just somewhat similar to "Sch" in Schroeder.

Any links to a definition for
\( \chi \) and \( \chi^{-1} \)

Ya, for example in the thread regular slog.
Literature is: Szekeres "Regular iteration of real and complex functions."

[sheldonison]

....
Unfortunately the convergence gets quite bad for \( b \) approaching \( e^{1/e} \), so I could not really check numerically.
On the other hand Walker describes also two solutions for \( b=e^{1/e} \) 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 \( e^{1/e} \).
....
in the thread regular slog.
Literature is: Szekeres "Regular iteration of real and complex functions."

Kouznetsov has graphs of the lower super exponential for \( b=e^{1/e} \) 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 \( b>e^{1/e} \), the function exponentially decays to its limiting value in the complex plane at +/- i \( \infty \). This is probably also true for the upper super exponential for \( b=e^{1/e} \), as the value at the real axis increases ...

[tommy1729]

As it is well-known we have for \( b<e^{1/e} \)
the regular superexponential at the lower fixed point.

This can be obtained by computing the Schroeder function at the fixed point \( a \) of \( F(x)=b^x \).

More precisely we set
\( G(x)=F(x+a)-a = b^{x+a}-a=b^a b^x -a = a b^x -a = a(b^x-1) \)
This is a function with fixed point at 0, it is the function \( F \) shifted that its fixed point is at 0.

We compute the Schroeder function \( \chi \) of \( G \), i.e. the solution of:
\( \chi(G(x))=c\chi(x) \) where \( c=G'(0)=a\ln(b)=\ln(b^a)=\ln(a) \).
This has a unique analytic solution with \( \chi'(0)=1 \).

Then we get the super exponential by
\( \operatorname{sexp}_b(t)=a+\chi^{-1}(c^x \chi(y_0) \)
\( y_0 \) is adjusted such that
\( 1=\operatorname{sexp}_b(0)=a+\chi^{-1}(\chi(y_0))=a+y_0 \)
i.e. \( y_0=1-a \).

This procedure can be applied to any fixed point \( a \) of \( b^x \).
The normal regular superexponential is obtained by applying it to the lower fixed point.

Now the upper regular superexponential \( \operatorname{usexp} \) is the one obtained at the upper fixed point of \( b^x \).
For this function we have however always \( \operatorname{usexp}(x)>a \),
so the condition \( \operatorname{usexp}(0)=1 \) can not be met.
Instead we normalize it by \( \operatorname{usexp}(0)=a+1 \), which gives the formula:
\( \operatorname{usexp}_b(t)=a+\chi^{-1}\left(\ln(a)^x \chi(1)\right) \)

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 \( x<-2 \).

It is entire because the inverse Schroeder function \( \chi^{-1} \) is entire, it can be continued from an initial small disk of radius r around 0 By the equation
\( \chi^{-1}(c^n x)=G^{[n]}(\chi(x)) \)
We know that \( c>1 \) thatswhy we cover the whole complex plane with \( c^nx \), \( x \) from the initial disc around 0, and we know that \( G^{[n}] \) is entire.

Here are some pictures of \( \operatorname{sexp} \) that are computed via the regular schroeder function as powerseries for our beloved base \( b=\sqrt{2} \), \( a=2,4 \):

[attachment=467]

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

nice post.

thanks.


regards

tommy1729

[bo198214]

Kouznetsov has graphs of the lower super exponential for \( b=e^{1/e} \) 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 \( b\uparrow e^{1/e} \) converges pointwise to the constant function \( e \) (which of course also a solution of \( f(x+1)=\left(e^{1/e}\right)^{f(x)} \)).

[sheldonison]

As it is well-known we have for \( b<e^{1/e} \)
the regular superexponential at the lower fixed point.

This can be obtained by computing the Schroeder function at the fixed point \( a \) of \( F(x)=b^x \).
.....
Now the upper regular superexponential \( \operatorname{usexp} \) is the one obtained at the upper fixed point of \( b^x \).
For this function we have however always \( \operatorname{usexp}(x)>a \),
so the condition \( \operatorname{usexp}(0)=1 \) can not be met.
Instead we normalize it by \( \operatorname{usexp}(0)=a+1 \), which gives the formula:
\( \operatorname{usexp}_b(t)=a+\chi^{-1}\left(\ln(a)^x \chi(1)\right) \)

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