Re: Tetration again!

On Dec 18, 12:02 am, mike3 <mike4...@xxxxxxxxx> wrote:
On Dec 17, 10:14 pm, Gottfried Helms <he...@xxxxxxxxxxxxx> wrote:





Am 18.12.2007 01:41 schrieb lwal...@xxxxxxxxx:

On Dec 17, 1:15 pm, Gottfried Helms <he...@xxxxxxxxxxxxx> wrote:
Am 17.12.2007 21:24 schrieb mike3:

I had a wild idea. Since you could differentiate tetration like that,
what happens if you try to expand it as a power series of some sort?
Esp. with the case b = e. Also, what happens when you try to find the
limiting case as the degree of approximation grows larger and
larger...?
I have, based on some conjecture, one layout for such a powerseries
in x, if you assume tetration

x if h=0
Tb(x,h) = b^x if h=1
b^Tb(x,h-1) if h>1

then Tb(x,h) can be expressed as a powerseries

Tb(x,h) = a0 + a1 x + a2 x^2 + ...

Very interesting. Sometimes I wondered whether tetration
has a simple power series.

Well, it seems to be not so simple... ;-)
We have three parameters: x (Top-exponent or initial value)
b (base) and h (height)
The b and h are "in" the a_k. Each a_k is a polynomial
in b and h - well, if we fix x to a certain value (x=1),
we may express this as a powerseries in h or in b.
So, btw, to discuss derivatives as Mike tries, needs to
explicate, which is the derivative-variable.

I was referring to the tower -- not the base, sorry if I
did not make this clear. I had been considering the
simplified case of the "tetrational function" -- that is,
the tetrational analogue of the exponential function,
namely that to some "easy" base that might make the work
simpler, say base e. Then the tetrational function is
tet(x) = ^x e. Ex. tet(1) = e, tet(2) = e^e, tet(3) = e^(e^e),
etc. Or perhaps, as you mentioned in the rest of the post,
the square root of two. Then try to find the most
natural possible extension in that base.

<snip>

PS. "Tower" here (as opposed to "base") is what you call
"height" -- I'm using the term "Tower" by analogy with
"power" -- (T)etrational p(OWER) -> TOWER, and as a pun.
.

Relevant Pages

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  • Re: Tetration again!
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