tommy's singularity theorem and connection to kneser and gaussian method
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Let K(s) = exp(K(s-1)) be the Kneser solution.

Let G(s) = exp(G(s-1)) be the gaussian method tetration.

Conjecture : K(s + a(s)) = G(s) , G(s + b(s)) = K(s) , where a(s) and b(s) are one-periodic analytic functions and one of a(s),b(s) is entire.

Lemma 1 : both K(s) and G(s) are analytic solutions.

Lemma 2 : from lemma 1 , there exists analytic periodic functions c(s),d(s) such that K(s + c(s)) = G(s) , G(s + d(s)) = K(s).

( ofcourse s + c(s) and s + d(s) are functional inverses )

Lemma 3 : G(s) is analytic where erf(s) is close to 1. ( triangle or sector )

Lemma 4 : tommy's singularity theorem :

Let tet(s) be analytic tetration such that when tet(s) is defined , so is tet(s + r) for real r >= 0.

Let tet(s) have a singularity at s = z and tet(s+1) has no singularity at s = z.

by the functional equation this implies that tet(z) is a logaritmic singularity.

It follows by induction :

if tet(s) has a singularity at s = z that is not a logaritmic ( ln or ln ln or ln ln ln or ... ) then tet(s+n) is also a singarity for all integer n > 0... or any integer n actually.

therefore , for any analytic tetration all non log-type singularities are 1 periodic !

Lemma 5 : IF tet(s) has non-log-type singularities then tet(s) = K(s + sing(s)) where sing is a 1 periodic function with singularities.

It seems to follow that 

lemma 6 : IF  tet(s) has no non-log-type singularities then tet(s) = K(s + theta(s)) where theta is a 1 periodic function without singularities.

the harder thing is to exclude log-type singul from theta(s).

Assuming those log-type are excluded from theta(s) in lemma 6 ;

lemma 7 : ... 

Since G(s) has no periodic singularities ( the triangle where erf converges fast to 1 forbids it ) , it follows that 

G(s) = K(s + theta(s)) for entire theta(s).

( again : the harder thing is to exclude log-type singul from theta(s) , the starting conjecture is a bit weaker such that inv( s + theta(s) ) can be entire ... and i assume that makes s + theta(s) have the log-type sing then.

if s + theta(s) really is entire then i assume inv( s + theta(s) ) has log-type singularities and branches due to s + theta(s) being flat ( derivat = 0 ) ... it seems to follow from the above that those " flat branches " must be log-type sing as well ...

THE REAL HARD PART is to exclude that both s + theta(s) and inv* have both log-type sing ! )

- more or less - QED

regards

tommy1729
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tommy's singularity theorem and connection to kneser and gaussian method - by tommy1729 - 09/18/2021, 12:06 PM

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