Re: Tetration again!

On Dec 19, 6:42 pm, Gottfried Helms <he...@xxxxxxxxxxxxx> wrote:
Am 19.12.2007 22:00 schrieb mike3:

On Dec 19, 9:17 am, Gottfried Helms <he...@xxxxxxxxxxxxx> wrote:
Am 19.12.2007 09:55 schrieb mike3:
<snip>

This is an interesting method. I'm curious, what happens
if you were to, just for fun, plug in a _complex_ number
for the tower ("height" as you call it)? Say, i? Would
the series still converge to something? If it does, is that
important and meaningful?

Also, what does a graph of ^x e look like, under this
method, on, say, x = -2 to 2? Is this extension
C^oo (infinitely differentiable, ie. "smooth"), at least
for the base e?

Hmm, there is a free math-program, Pari/GP, with which
you can try such things. I've also written a graphical
user-interface for it (windows XP), search for "Pari-tty"
If you want to work seriously with it, I could provide
you with a set of basic matrix-routines for a start.



In addition, at what tower does the function ^x e equal
the square tetraroot of e, approximately 1.7632228343519,
using your method? It obviously doesn't at 1/2.

Also, what about non-e bases? Note that for small
bases, say, base 1.5, the integer-tetration graph slows
in growth at a point before taking off? (Presumably
it does this infinitely often since that feature i
carried over by the relationship ^x b = b^(^(x-1) b),
but the numbers become too immense to tell.) Does your
method give a similar graph?

The polynomials become very complex - each term is then
described by a matrix: small for the leading terms, growing
size with higher index, apparently with binomial growing
order. The Mlog-function is then no more useful; I've a
symbolic description of the Eigensystem instead - but
there must still be some small error, since it computes fractional
iterates of complex fixpoints (complex bases or bases>e^(1/e))
not correctly- for integer heights it works fine... I seem
to have errors related to the multivaluedness of logarithm
in the definition/computing procedures.


This is odd. The series you give does not have logs in
it, just the parameters x and h, so how does multivaluedness
of the logarithm affect this? One can just plug "i" in for
h and see what comes out, no? Using the first 5 terms you
posted, setting x = 1, h = i, I get U(1, i) ~ 0.9 + 0.4i.
I don't know how accurate that is, since I'm not quite sure
how to generate additional terms.

Also, U(1, 1) with the terms you posted gives 1.71828...
not ^1 e = e = 2.71828... what gives?
.