Could be tetration if this integral converges

[tommy1729]

I totally agree. Although I added a minus sign to it, which I assume was a typo.

So lets think about .

Since the Taylor coefficients of decay EXTREMELY FAST , I consider this as a function that is well approximated by a polynomial for a long time.
( many remainder theorems for Taylor series imply this )

This means the main behaviour of this is like where n increases slowly with x.

This implies that is not bounded by a polynomial and also that = 0 infinitely often.

Therefore the integral diverges.

Even if we consider taking the limit of x going to +oo as the limit of the sequence x_i with = 0.

regards

tommy1729

yes yes, I'm quite aware it diverges. That's only another trick we need to come up with to handle that. I have a few but I need to look deeper into the laplace transform.

I assumed you were aware of it. But some readers might not have been convinced.
With that in the back of my mind, I felt the neccessity to reply.

Its pretty hard to combine the properties of convergeance and the functional equation with fractional calculus and integral transforms ... or so it seems.

Maybe a bit of topic but finding an approximation to in terms of an integral transform seems to be closer to a solution due to the recent work and talk of myself and sheldon.



where is bounded by a constant above and a constant below , and x > 27.

(And the factorial is computed with the gamma function ofcourse )

with approximation I mean that they have the same " growth rate ".

regards

tommy1729