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Can sexp(z) be periodic ??
#1
I wonder if sexp(z) can be periodic.

In particular 2pi i periodic.

A search for period(ic) gives too many results.
SO forgive me if this has been asked/answered before.

If I recall correctly a pseudoperiod of 2 pi i / L is known to be possible ( L is a fixed point of exp ) because of f(z) = lim exp^[n](L^(z-n)) and similar limits.

Im unaware of limit formula's that give a periodic solution.

Maybe pentation helps us out , although I want to avoid adding this function.

Maybe the system of equations is overdetermined ??

f(0) = 1
f(z+a) = f(z)
f(z+1) = exp(f(z))

On the other hand maybe it has uniqueness ?

I think it has uniqueness if it has existance.

Reason is f(z+theta(z)) = f(z+a+theta(z+a)) Implies that

theta(z) is both '1' and 'a' periodic HENCE double periodic ;

A nonconstant theta can thus not be entire here !

COMBINE THAT WITH THE RIEMANN MAPPING THEOREM AND

nonconstant theta(z) cannot exist !!!

( I think )

regards

tommy1729
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#2
(01/12/2015, 12:46 AM)tommy1729 Wrote: I wonder if sexp(z) can be periodic.

In particular 2pi i periodic....
tommy1729

L=0.318131505204764 + 1.33723570143069i, and one can develop the standard Schroder equations about the fixed points. At the fixed point, , where is the fixed point multiplier, since

The definition of the formal Schroder equation, which leads to a formal Taylor series is



So then


The super function is also entire. Of course, the Schroder function of 0,1,e,e^e, are all singularities... so this function needs a lot of work to become the real valued sexp(z) we use for Tetration, but it is the starting point...

- Sheldon
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#3
(01/12/2015, 01:54 AM)sheldonison Wrote:
(01/12/2015, 12:46 AM)tommy1729 Wrote: I wonder if sexp(z) can be periodic.

In particular 2pi i periodic....
tommy1729

L=0.318131505204764 + 1.33723570143069i, and one can develop the standard Schroder equations about the fixed points. At the fixed point, , where is the fixed point multiplier, since

The definition of the formal Schroder equation, which leads to a formal Taylor series is



So then


The super function is also entire. Of course, the Schroder function of 0,1,e,e^e, are all singularities... so this function needs a lot of work to become the real valued sexp(z) we use for Tetration, but it is the starting point...

But the Schroder function does not give a real-analytic sexp.

I prefer not to " abuse " notation.





So then


Im intrested in both periodic sexp's ; both real-analytic and not real-analytic.

I was thinking about other limits forms , for instance including terms like exp(z) to " force " periodicity , but I have convergeance issues that cannot be solved by analytic continuation.

regards

tomm1729
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