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I would like to talk about the conjectured phenomenon of an analytic solution sexp that is " pseudo2periodic " in a big part of the complex plane.

Basicly this is a talk about how sexp could look like according to me.
The idea is related to many previous ideas such as the boundedness conjecture and the development of " fake function theory ".
" Fake double periodic " is perhaps an alternative name for speudo2periodic.

But enough intro.

Lets assume a function f1(z) has a single fundamental fixpoint F, then its superfunction f2(z) in the complex plane is given by the limit equation :

$\operatorname{SuperFunction}(z) = \lim_{n \to \infty}
\operatorname{f1}^{[n]}(F^{z-n})$

Since F^z is periodic with period = 2pi i / ln(F), then the superfunction f2(z) is also periodic with that same period.

It requires more formal thinking but it seems logical to say that if a function has a single fundamental conjugate fixpoint pair F,F* and the derivatives at those points are also F,F* resp. then in the upper resp lower plane we get the 2 pseudoperiods 2pi i / ln(F) and 2pi i / ln(F*).

This leads to the idea that sexp(z) in a part (explained later) of the upper plane has a pseudoperiod of P1 = 2 pi i / L.
(L is the upper fundamental fixpoint of exp(z) ).

As the title suggest I must have another pseudoperiod , because pseudo2periodic means 2 pseudoperiods.

This would give our sexp a tiling type of structure what is very informative considering the complexity of the function.

It is well known that an entire double periodic function must be constant.
But this function sexp(z) is neither entire nor double periodic so those related theorems do not apply. ( poles , singularities , constants , ... )

Let L = A + B i.
Arg(L) = Arctan(Y/X) = theta

The second pseudoperiodic number is P2 = 2 pi i / theta i.

This can be reduced to P2 = 2 pi / Arctan(Y/X) and thus this is a real number.

The idea came from " going around in a circle ".

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This second pseudoperiod might however not be a fixed number , it is P2 + sin© for some real C and this depends on the theta wave , in other words the type of sexp used.
It is Always real though and sin© is Always a small number.
So we still have pseudo2periodicity.

I think that sin© = 0 though and Im thinking about a proof.
Update : found a proof.

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Now the issue of " where " sexp is psuedoperiodic is resolved as follows :

We solve the problem for the upper complex plane , similar for the lower.

Consider Im(z) > 2 pi.
Draw a line for Im(z) = 2.
Draw the line with angle arg(P1) going through the origin [point (0,0)] on the complex plane.
Now these 2 lines intersect in one point.

We now have 4 regions.

Now the pseudo2periodicity takes place in the upperleft region of the intersection point.

Going very far from away from the intersection point in the upperleft region means going to the fixpoint L with exponential speed.
( For those who understand this well , this all implies the speed can be computed when given how exactly we move away )

1) the boundedness conjecture.
( or is it implied ? )

2) analyticity
( I think for the entire upper plane ? or is it implied ? )

3) No real number in the consider upperleft region.

The third one is probably the trickiest.

I ve been thinking how it relates to sexp ' (z) = 0 but that got complicated and I stopped for now.

Let me know what you think.

regards

tommy1729