Re: continuous iteration

On May 25, 8:44 pm, galathaea <galath...@xxxxxxxxx> wrote:
On May 22, 11:57 pm, Tonico <Tonic...@xxxxxxxxx> wrote:





On May 23, 2:48 am, galathaea <galath...@xxxxxxxxx> wrote:
sometimes
  as in the case of gamma
you need to add conditions for uniqueness
(like log convexity does with bohr-mollerup)
(and it seems like in your last message

Tetration seems to be a pretty edgy subject,
and since it apparently uses the well known
Gamma function, poor tommy thinks (so to speak)
that if someone doesn't know about tetration
then he also doesn't know about the Gamma
function...:)

 you may not be noticing gamma itself is a continuous iterate)

there has never been anything "edgy" about tetration

it has a long history

******************************************************

I don't think so: I think it's hugely edgy, but it doesn't really
matter: you may as well not consider it so.
About gamma being continuous iterate: do you mean that it fulfills

G(s+1) = s*G(s), or what? I never heard it called that way, but that
jsut could be my bad. I hear id called "the factorial function", we
even proved in an advanced calculus course that G is the unique real
function which fulfills:
1) G a log-convex function;
2) G(1) = 1
3) G(s+1) = sG(s)

the factorial of n can be seen as
"multiply the n numbers from 1 to n together"

1.2.3...n

which can be written in the rising pochhammer

(1)
   n

now
  much as i demonstrated for changing
  "sum the n numbers f(.)..."
there is a way to make sense of "continuous products"

again
  much as with sums
the idea is to extend the iteration infinitely
with a shifted part that normally cancels
  but some extra care to avoid infinities is necessary

write

       (1)      (1) (1+n)
          x+n      n     x
(1)  = ------ = ----------
   n   (1+x)      (1+x)
            n          n

now
the leading term of

(1+n)
     x

    x
is n

so the limit as n->oo of the above
  is the same limit as

       (1)    x
          n  n
 lim   --------
n->oo   (1+x)
             n

which is defined for all x
and is one way euler defined

__
| '
|  (1 + x)

           ..

so
now i've shown you ways of making continuous
  iterative processes for sums and products

do you see how iterative exponentiation
  can also be so generalised?

several different ways have been proposed
but there is one approach that follows this same idea

can you guess it now?

********************************************************

I sincerely thank you for trying to make some sense for me out of all
the above, but I still can't see it: you began by saying that

n! = (1)~ n -- meaning (1) is kind of power to the left of n --.

Then you write (1)~(x + n), which I can udnerstand as being (x + n)!
generalized, most like the infinite product in (a+b)^1/2 as series,
trying to generalize Newton.

But then you write (1+x)~ n, and I just can't understand what this
mean: factorial from 1+x to n or what..??

Not that iteration or whatever has anything to do with the solution of
that darn old hoax, but it would nevertheless be nice to understand
something new in maths.

Regards
Tonio
.

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