Real-analytic tetration uniqueness criterion?

[tommy1729]

For tetration at the real axis, the nearest singularity is at x=-2, and there are no other singularities to the right of that anywhere in the complex plane.

Well that depends on what type of tetration we use.
Kneser seems to have this property.
But so do many theta variations of Kneser. ( the analytic theta's )
Notice by adding the condition bounded in a strip we get an old uniqueness criterion.

So, for all points on the real axis, the derivatives must eventually be governed by log(x+2), for large enough derivatives! And the even derivatives of log(x+2) are all negative. So I think that is a proof that for any point on the real axis, no matter how large, eventually the (2n)th derivative must be negative for large enough values of n. However, all of the odd derivatives are always positive.

Hmm why not all of the odd derivatives are positive FOR LARGE ENOUGH values of n ??

Because if so , then it follows from

sexp has a log singularity at -2

=> f(x) = sexp(x) - log(x+2) is analytic for x > -3.
now when expanded at y > -2 then

f(x) has a larger radius than log(x+2)
Thus the derivatives of log(x+2) expanded at y must eventually dominate the derivatives of f(x) expanded at y.

SO expansion at y > -2 =>
sexp(x) = log(x+2) + f(x)
DOMIN sexp(x) = DOMIN(log(x+2),f(x)) = DOMIN log(x+2).

(QED)

But again : Hmm why not all of the odd derivatives are positive FOR LARGE ENOUGH values of n ??

AND what happens with sexp(z+theta(z)) ?

is f_2(z) = sexp(z+theta(z)) - log(z+theta(z)+2) still analytic for Re(z) > -3 ???

If sexp(z+theta(z)) is analytic for Re(z)>-2 then we seem to require

log(z+theta(z)+2) ~ log(z+2)

However theta(z) = 0 may have many solutions ... SO IN GENERAL it is NOT true ! We have a bad approximation and thus did not remove the singularity without adding another one.

Notice : DOMIN sexp(z+theta(z)) = DOMIN(f_2(z),log(z+theta(z)+2))
Together with theta is analytic and 1 periodic with fixpoint 0 , this gives quite an intresting structure.

Lets assume theta(z) = 0 only for one value z (= 0).

then theta(z) is no longer periodic !

contradiction.
But however no issue since these values are far from z = 0.

SO we need to look again at :

log(z+theta(z)+2) ~ log(z+2).

log(z+theta(z)+2) - log(z+2) = g(z).

Now Im convinced g'(0) = 1 is necc otherwise the approximation is not good.

Hence this requires theta ' (0) = 0.
IF NOT the singularity is not properly removed right ?

And ofcourse if the singularities are not properly removed the Domination is not clear.

This was also my motivation for having theta ' (0) = 0 in the past.
( such as sin^2(x)/K as theta where the derivative IS 0 , not just in the limit K -> oo )

Needs more investigation ...

Now the other side of the conjecture is that for tet(z+theta(z)/k), eventually for large enough odd derivatives misbehave, and the odd derivatives are no longer all positive. I believe a possible avenue to proving this is to imagine a circle from (-1) to z0, where we are looking at the odd derivatives at a point (z0/2-1/2). Without the theta perturbation, the tetration function on the boundary of that circle has its maximum positive value at z0.

See the comments above !

Also why would the maximum value be at z0 ?

This is not in general true for polynomials UNLESS all derivatives are nonnegative.
But here we have some negatives. Well even that depends on parameters and conjectures.

I do not see then why it would be the case here.
But maybe it requires just a bit more explication.

This is not a zeta function afterall !
( for the Riemann zeta we have that for Re(z) > 1 any circle that lies to the right of it , with real center A , reaches its max at the largest real on the real line touching the circle. This follows easily from the exp function. )
Also some fake function of ln(z+w(z)) seems to be a counterexample for some 1 periodic theta function w(z) that is choosen wisely ( to have signs for the nth derivative the way we want ).
But fake function theory is overkill here.

... The algorithm is based on the algorithm in post#16, I used for the Taylor series of the entire half sexp approximation thread; it will take me a little while to write up the equations for this algorithm.

Fake function theory again !

Lots of work to do again.

regards

tommy1729

" thruth is that what does not go away when you stop believing in it "