03/29/2009, 11:23 AM
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
=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
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
=a+\chi^{-1}(c^x \chi(y_0))
is adjusted such that
=a+\chi^{-1}(\chi(y_0))=a+y_0)
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:
=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
.
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
=G^{[n]}(\chi(x)))
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:
the regular superexponential at the lower fixed point.
This can be obtained by computing the Schroeder function at the fixed point
More precisely we set
This is a function with fixed point at 0, it is the function
We compute the Schroeder function
This has a unique analytic solution with
Then we get the super exponential by
i.e.
This procedure can be applied to any fixed point
The normal regular superexponential is obtained by applying it to the lower fixed point.
Now the upper regular superexponential
For this function we have however always
so the condition
Instead we normalize it by
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
We know that
Here are some pictures of
and here the upper super exponential base 2 alone: