08/30/2010, 03:09 AM
(This post was last modified: 08/30/2010, 03:33 AM by sheldonison.)

(08/28/2010, 11:21 PM)tommy1729 Wrote: that seems efficient and intresting.Thanks Tommy! I assume it has been considered, and probably calculated before. I think Kneser developed the complex periodic regular tetration for base e, and probably would've generated the coefficients. But I haven't seen them before. Perhaps Henryk (or someone else) could comment???

in fact i doubt it hasnt been considered before ?

I figured out the closed form equation for a couple more terms, and I have an equation that should generate the other terms, but I'm still working it, literally as I write this post!

What I did is start with the equation:

and set it equal to the equation

Continuing, there is a bit of trickery in this step to keep the equations in terms of , instead of in terms of . Notice that .

This becomes a product, with and

The goal is to get an equation in terms of on both sides of the equation. Then I had a breakthrough, while I was typing this post!!!! The breakthrough is to set , and rewrite all of the equations in terms of y! This wraps the 2Pi*I/L cyclic Fourier series around the unit circle, as an analytic function in terms of y, which greatly simplifies the equations, and also helps to justify the equations.

The next step is to expand the individual Tayler series for the , and multiply them all together (which gets a little messy, but remember a0=L and a1=1), and finally equate the terms in on the left hand side equation with those on the right hand side equation, and solve for the individual coefficients. Anyway, the equations match the numerical results.

I'll fill in the Tayler series substitution next time; this post is already much more detailed then I thought it was going to be! I figured a lot of this out as I typed this post!

- Sheldon