05/27/2011, 09:36 AM
(This post was last modified: 06/02/2011, 08:09 PM by sheldonison.)

Continuing with the conjecture, , here are some more graphs, and then some comments. cheta(z) is the upper entire superfunciton of eta.

from z=-1+i to z=1+i, which I previously posted, where theta(z) is well behaved, and decaying as imag(z) increases.

Next, lets regenerate this graph with , which is closer to the real axis, and closer to the singularity at ineger values of z.

To get closer to the real axis, we need to switch to a graph of the contour of , by graphing the contour of

. The range if this graph is identical to the previous graph, from z=-1+0.001i to 1+0.001i. Only the real part of the constant k has been included, so the imag(z) values represent the contour values of cheta(z).

This last cheta(z) contour plot is a graph of the Kneser mapping contour for cheta(z), at the real axis itself, calculating the contour , with a range of z from to . This is equivalent to the range of z from to , where delta_a and delta_b are very small. Near -2, . I think that the ... messy arithmetic. The graph has been modified so that it matches the range from -1 to +1 from above. For base e, I have observed that the graph of the Knser mapping contour continues becoming more and more complex, as we superexponentially approach the singularity. Similar complexity may occur for base eta. I may post more in the future.

Tommy wrote:

Hey Tommy. There are some restrictions. These Kneser/Riemann mappings involve

1) a theta(z) function quickly decaying to zero at +imag infinity,

and

2) a resulting function with singularities at the integer values, where the singularities results in a Schwarz reflection, which allows the function to be defined for imag(z)<0.

We already have the regular superfunction for base e, as an example, which is Kneser mapped to produced sexp_e(z). This is another example, where the upper/entire superfunction, cheta(z) is Kneser mapped to produce sexp_eta(z).

These two restrictions, limit the kinds of functions involved. For tetration, this works for bases>eta, using the standard Kneser mapping, and for base cheta, as is conjectured.

For bases<eta, other theta(z) mappings are possible. I made an entire post about them last year, where I discussed base 2. http://math.eretrandre.org/tetrationforu...hp?tid=515

I have to refresh my memory on what I've posted, but I also derived a new different tetration solution for each base less than eta, using a Kneser mapping.

- Sheldon

from z=-1+i to z=1+i, which I previously posted, where theta(z) is well behaved, and decaying as imag(z) increases.

Next, lets regenerate this graph with , which is closer to the real axis, and closer to the singularity at ineger values of z.

To get closer to the real axis, we need to switch to a graph of the contour of , by graphing the contour of

. The range if this graph is identical to the previous graph, from z=-1+0.001i to 1+0.001i. Only the real part of the constant k has been included, so the imag(z) values represent the contour values of cheta(z).

This last cheta(z) contour plot is a graph of the Kneser mapping contour for cheta(z), at the real axis itself, calculating the contour , with a range of z from to . This is equivalent to the range of z from to , where delta_a and delta_b are very small. Near -2, . I think that the ... messy arithmetic. The graph has been modified so that it matches the range from -1 to +1 from above. For base e, I have observed that the graph of the Knser mapping contour continues becoming more and more complex, as we superexponentially approach the singularity. Similar complexity may occur for base eta. I may post more in the future.

Tommy wrote:

Quote:another question is : how many superfunctions can a function have ?

in this thread we have a lower and upper superfunction.

but when considering complex numbers and non-real fixpoints and general analytic functions , i wonder about how many superfunctions one can have and how to determine them.

Hey Tommy. There are some restrictions. These Kneser/Riemann mappings involve

1) a theta(z) function quickly decaying to zero at +imag infinity,

and

2) a resulting function with singularities at the integer values, where the singularities results in a Schwarz reflection, which allows the function to be defined for imag(z)<0.

We already have the regular superfunction for base e, as an example, which is Kneser mapped to produced sexp_e(z). This is another example, where the upper/entire superfunction, cheta(z) is Kneser mapped to produce sexp_eta(z).

These two restrictions, limit the kinds of functions involved. For tetration, this works for bases>eta, using the standard Kneser mapping, and for base cheta, as is conjectured.

For bases<eta, other theta(z) mappings are possible. I made an entire post about them last year, where I discussed base 2. http://math.eretrandre.org/tetrationforu...hp?tid=515

I have to refresh my memory on what I've posted, but I also derived a new different tetration solution for each base less than eta, using a Kneser mapping.

- Sheldon