I propose an analytic tetration here.
One very similar to other ideas from this year.
Imo very natural and just requiring alot of simple computions. I dare say brute force.
Consider for real k > 5 and Re(s) > 0 :
\( A_k(s)=exp(A_k(s-1) - exp(-ks)) \)
We can solve this by infinite composition : \( A_k(s)=exp( - exp(-ks) + exp( - exp(-k(s-1)) + ...) \)
( yes to be formal we need a " z parameter " too, but that is a detail )
Now consider the functions \( A_5(s),A_6(s),A_7(s),... \)
The limit k to +oo gives us A_oo(s) = A(s).
Now A(s) satisfies for Re(s) > 0 :
A(0) = exp( -exp(-oo) + exp(-oo + 0)) = exp(0 + exp(-oo)) = exp(0) = 1.
and
A(s+1) = exp(A(s) + 0) = exp(A(s))
So assuming the limit exists this is analytic tetration on the halfplane Re(s) > 1.
By analytic continuation ( taking log's ) this is analytic tetration !!
In fact I think it has no singularities for Re(s) > 1 making it perhaps satisfy a uniqueness condition ??
The limit is designed to ( when it exists ) grow slowly on the complex plane (in the imaginary direction ) for Re(s) > 1 therefore probably remaining analytic.
We could in principle even approximate this in say excel.
regards
Tom Marcel Raes
tommy1729
One very similar to other ideas from this year.
Imo very natural and just requiring alot of simple computions. I dare say brute force.
Consider for real k > 5 and Re(s) > 0 :
\( A_k(s)=exp(A_k(s-1) - exp(-ks)) \)
We can solve this by infinite composition : \( A_k(s)=exp( - exp(-ks) + exp( - exp(-k(s-1)) + ...) \)
( yes to be formal we need a " z parameter " too, but that is a detail )
Now consider the functions \( A_5(s),A_6(s),A_7(s),... \)
The limit k to +oo gives us A_oo(s) = A(s).
Now A(s) satisfies for Re(s) > 0 :
A(0) = exp( -exp(-oo) + exp(-oo + 0)) = exp(0 + exp(-oo)) = exp(0) = 1.
and
A(s+1) = exp(A(s) + 0) = exp(A(s))
So assuming the limit exists this is analytic tetration on the halfplane Re(s) > 1.
By analytic continuation ( taking log's ) this is analytic tetration !!
In fact I think it has no singularities for Re(s) > 1 making it perhaps satisfy a uniqueness condition ??
The limit is designed to ( when it exists ) grow slowly on the complex plane (in the imaginary direction ) for Re(s) > 1 therefore probably remaining analytic.
We could in principle even approximate this in say excel.
regards
Tom Marcel Raes
tommy1729
