Could be tetration if this integral converges

[JmsNxn]

And by the asymptotics of as we get!

I believe this is where the problem lies. As I mentioned, the reciprocal tetrational is going to be unbounded. I believe it is also possible with the topological-transitivity thing I mentioned to show that it will also take on almost every complex value infinitely often, on . So as the arc cuts through that highly ill-behaved region, there seems no reason to assume this integral must converge to as .

YES yes yes yes. Thank you. I don't know very much on how tetration behaves imaginarily. I just meant IF! it satisfied that exponential bound we'd be good. implies it cannot satisfy that exponential bound or an even stronger bound (we can apply this method to more functions not satisfying an exponential bound but it requires a more heavy treatment of the gamma function and a proof of

If what typically happens when the function does not have an exponential decay happens then we usually use some asymptotic analysis or an expansion:

where is typically meromorphic on all of .


And on your point about uniqueness. I was being very brief but giving an over view of how we may be able to qualify uniqueness. In technical terms, "it's the only function that in the inverse mellin transform produces an entire function f that is Weyl differintegrable on the right half plane "

However after seeing what you just posted I have to draw the same conclusion as you. has no hope of converging in a mellin transform. Which is what I pretty much figured. BUT! we're not out of the woods yet.


IF we can find some entire function such that for :



we are back in the game

OR IF we can find some entire function such that for for and

we are back in the game.


By back in the game I mean I think I can provide an analytic expression for tetration. I'm just finishing the paper I'm working on at the moment and it contains a fair amount of what I'm talking about a lot more rigorously. I'll attach it once I know it's in it's final form. It shows what I am talking about more c learly when I am using fractional calculus on recursion.