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

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:

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: