Re: Tetration again!

Am 21.12.2007 22:29 schrieb mike3:

What would be really interesting is if this method applies
more generally to iterating functions continuously *and*
reproduces the first 3 hyper operators when applied
to their iterations, ex. for multiplication we
iterate f(x) = b + x, for exponentiation we iterate
f(x) = bx, and the results agree exactly with the
widely-accepted definitions.
I don't remember the post, but I wrote about his one
or two times.
Using matrices you may describe the operations + and *
the same way as I do it with the tetration-matrices (T-
and U-tetration), using powerseries.

For "+" operation you may use the pascal-matrix P,
which implements the binomial-theorem for powerseries:
V(x)~ * (P^b)^h ~ = V(x+b*h)
in scalar mode
x + (b + b + b + ...) //h-fold occurence of b
= x + b*h
with fractional or complex heights h.

For "*" -operation you just use a diagonal-matrix
containing consecutive powers of b, and

V(x)~ * dV(b)^h = V(x*b^h)~
in scalar
x * (b*b*b*...*b) // h-fold occurence of b
= x*b^h

again h can be complex here.

Gottfried



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Gottfried Helms, Kassel
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