(05/05/2014, 02:11 AM)mike3 Wrote:I'll note firstly that(05/04/2014, 09:06 PM)JmsNxn Wrote:

So is this supposed to approximate tetration if is small? As if so, then it doesn't seem to be working for me. If I take and the integral upper bound at 2000, I get as ~443444.33873479713260158296678612894384. Clearly, that can't be right -- it should be between and (if this is supposed to reproduce the Kneser tetrational then it should be ~5.1880309584291901006085359610758671512). It gets worse the smaller you make -- i.e. it doesn't seem to converge. Also, picking values to put in that are near-natural numbers doesn't seem to work, either.

I accidentally added an extra negative. But that doesn't really affect convergence. I understand whats happening.

Hmm. That makes sense now that I think about it. I was hoping you could take lambda small but not too small and it wouldn't diverge too fast but because obviously diverges this doesn't happen. Maybe if you try I've done more research into this form of the operator so perhaps we can work with this one.

I do have a nice result that will work for these operators that is slightly off topic but is related to continuum sums. If is as before and :

Quite fantabulously if we get the really interesting result:

And of course:

I'm a little fuzzy on the following but if we use a fractional iteration of we can find a very nice interpolant of . Since it's all using fractional calculus I have an impressionistic vision of how a similar proof of recursion would go (picking a certain fractional iteration of I have in my mind and then using a similar contour integral technique).

I'm really intrigued by this idea but I think I have a more general result we need. I wonder if there exists a theorem in complex analysis on the following.

If is holomorphic on does there exist some holomorphic function holomorphic on such that:

for and

such that satisfies some conditions I'm not sure of yet. It cannot interpolate , it cannot interpolate the inverse and it cannot be a fair amount of obvious easy functions.