I hesitated to post this because its another set of crazy ideas.
But it might inspire you.
I talked about getting h(s) from f(s).
But ofcourse similar ideas are getting an f(s) from h(s).
Another idea is getting h from f ( or f from h ) also in infinite functional composition.
But how ?
Keeping these in mind.
A more logical estimate for h(s) for RE(s) < 1 might be
h(s) = T(s)
where
T(s) = (s+1) / (exp(-s) + 1)
Remember that f(s) usually has the property that for Re(s) going to - oo , f(s) is going to zero.
Hence f(h(s)) should be getting close to f(0).
T(s) more or less does that.
Another idea is that h(s) is actually the superfunction of T(s) !!
that might be a better estimate also for Re(s) being a large positive real.
In BOTH cases we wonder about iterations of T(s) or equivalently its super.
And the iterations of the super of T(s) OR the super of the super of T(s).
So where to start ?
Well the most logical would be the primary fixpoints of T(s).
But wait a minute.
T(s) = s has the same solution as exp(s) = s !!
I notice that not much has been said about what happens to the fixpoints of exp for f,g,h.
( for instance at + oo i ?? )
since T(s) = h(s) = s , we arrive at f(h(s)) = f(s) and more ;
f(s) = exp(f(s)) = exp(s) = s for that primary fixpoint of exp !
***
Better estimates remain important ofcourse ...
But lets consider further.
I do not think the superfunction of T(s) has ever been considered.
I find it interesting because T(s) is close the successor function.
And that inspired me to consider this as part of a hyperoperator family.
Zeration , ackermann that kind of ideas with T(s).
Therefore I considered the slight generalization :
T_v(s) = (s+v) / (exp(-s) + v).
T_e(s) is then an attractive idea.
What is T_v(s) a superfunction of ?
Is creating a super of T_v(s) simpler than for exp ?
I considered carlemann matrices but it did not illuminate me.
regards
tommy1729
But it might inspire you.
I talked about getting h(s) from f(s).
But ofcourse similar ideas are getting an f(s) from h(s).
Another idea is getting h from f ( or f from h ) also in infinite functional composition.
But how ?
Keeping these in mind.
A more logical estimate for h(s) for RE(s) < 1 might be
h(s) = T(s)
where
T(s) = (s+1) / (exp(-s) + 1)
Remember that f(s) usually has the property that for Re(s) going to - oo , f(s) is going to zero.
Hence f(h(s)) should be getting close to f(0).
T(s) more or less does that.
Another idea is that h(s) is actually the superfunction of T(s) !!
that might be a better estimate also for Re(s) being a large positive real.
In BOTH cases we wonder about iterations of T(s) or equivalently its super.
And the iterations of the super of T(s) OR the super of the super of T(s).
So where to start ?
Well the most logical would be the primary fixpoints of T(s).
But wait a minute.
T(s) = s has the same solution as exp(s) = s !!
I notice that not much has been said about what happens to the fixpoints of exp for f,g,h.
( for instance at + oo i ?? )
since T(s) = h(s) = s , we arrive at f(h(s)) = f(s) and more ;
f(s) = exp(f(s)) = exp(s) = s for that primary fixpoint of exp !
***
Better estimates remain important ofcourse ...
But lets consider further.
I do not think the superfunction of T(s) has ever been considered.
I find it interesting because T(s) is close the successor function.
And that inspired me to consider this as part of a hyperoperator family.
Zeration , ackermann that kind of ideas with T(s).
Therefore I considered the slight generalization :
T_v(s) = (s+v) / (exp(-s) + v).
T_e(s) is then an attractive idea.
What is T_v(s) a superfunction of ?
Is creating a super of T_v(s) simpler than for exp ?
I considered carlemann matrices but it did not illuminate me.
regards
tommy1729