(05/04/2014, 09:06 PM)JmsNxn Wrote: As a note on a similar technique you are applying Mike but trying to keep the vibe much more fractional calculus'y (since it's what I am familiar with) We will try the following function:

We want fairly small

We know that for because

So that .

This function should be smaller then tetration at natural values.

We would get the entire expression for by Lemma 3 of my paper:

Now F(z) will be susceptible to alot of the techniques I have in my belt involving fractional calculus. This Idea just popped into my head but I'm thinking working with a function like this will pull down the imaginary behaviour and pull down the real behaviour.

We also note that

. Which again will be more obvious if you look at the paper, but it basically follows because:

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.