Re: That darned tetration again!

On Jun 13, 12:45 pm, lwal...@xxxxxxxxx wrote:
On Jun 12, 3:01 pm, mike3 <mike4...@xxxxxxxxx> wrote:



I just recently remembered a bit of a discussion here I had a while
ago about if it
was possible to come up with a "reasonable" set of properties for a
tetration
function, then see if it was possible to prove a unique function
existed that
satisfies them.
The original list of properties was:
(Here, b and x are real numbers with b > 1 and x >= 1)
1. f(b, x) satisfies the recurrence relation f(b, x+1) = f(b, x).
2. f(b, 1) = b.
3. f(b, x) is infinitely differentiable everywhere (perfectly smooth)
in both x and b.
4. f(b, x) increases strictly monotonically with respect to x, as do
all it's derivatives with respect to x.
5. For every x, there exists a T such that f(f(b, T), x) = b. (ie.
tetraroot of b exists and can be extracted via tetration.)
After playing with some extensions of tetration, it appears that
4 and 5 were not good. Actually, 5 was good but "T" appeared
to be dependent on b, not just on x.

Properties 4 and 5 are rather interesting. But before
discussing them more, let's observe a few things more
about tetration.


Actually, I don't think they'll work. See below.

Let b=sqrt(2) and note the following:

sqrt(2)^^0 = 1
sqrt(2)^^1 = 1.414213562
sqrt(2)^^2 = 1.632526919
sqrt(2)^^3 = 1.760839556
sqrt(2)^^4 = 1.840910869
sqrt(2)^^5 = 1.892712697
sqrt(2)^^6 = 1.926999702
sqrt(2)^^7 = 1.950034774
sqrt(2)^^8 = 1.965664887
sqrt(2)^^9 = 1.976341754
sqrt(2)^^10 = 1.983668399
sqrt(2)^^20 = 1.999585623
sqrt(2)^^30 = 1.999989394
sqrt(2)^^40 = 1.999999728
sqrt(2)^^50 = 1.999999993

(Notice that above I'm using te notation b^^x where
mike3 uses f(b,x).)


Yes, I use f(b, x) because it's an unknown function. If it
satisfies those properties and is _unique_ in doing so, then
it seems "natural" to consider it an extension of tetration,
in which case we would define

^x b = f(b, x).

(or as you write it, b^^x = f(b, x).)

Note the convergence though with b = sqrt(2). That means
it's growth is _slowing down_. Which means that df/dx
will be _decreasing_, and so Property 4 on my original list
fails. Hence the revision in the new list of properties.

We see that lim x->infinity f(sqrt(2),x) isn't infinity,
as one might expect. Indeed, it is easy to prove that
lim x->infinity f(sqrt(2),x) = 2.


Yes, that's right.

But of course lim x->infinity f(2,x) really is infinity,
which may lead one to wonder there exists a real number
eta such that for all 1 < b < eta, lim x->infinity f(b,x)
converges, and for all b > eta it diverges.

Such an eta does exist. Indeed, one can prove that eta
has the value e^(1/e). Furthermore, we can show that
lim x->infinity f(eta,x) = e.

So now we may revisit mike3's property 5:

5. For every x, there exists a T such that f(f(b, T), x) = b.

Since mike3 states that he desires T to depend solely on
x, let us emphasize this by writing T = T(x).

Ioannis Galidakis, who has already posted in this thread,
first speculated that T(x) = 1/x, in analogy with the
operation of exponentiation. At the following link, he
provides a graph for b = e, on the interval [0,1]:

http://ioannis.virtualcomposer2000.com/math/exponents7.html

As one can see, the function doesn't appear to be smooth
(mike3's Property 3) at x = 0 or x = 1. Indeed, we can
show that the function isn't even _continuous_, much less
_smooth_, at x = 0. Why not? It's because f(e,0) = 1, but
what is lim x->0+ f(e,x)? It would have to be the same as
lim x->infinity tetraroot(e,x) -- but notice that every
tetraroot of e must be greater than eta, since we have
lim x->infinity f(eta,x) = e. In fact, we can show that
lim x->0+ f(e,x) = eta /= f(e,0) = 1. Therefore, f is
not continuous!

So this rules out T(x) = 1/x, but it doesn't rule out
the existence of a function T(x). Indeed, all we have
proved is that lim x->infinity T(x) /= 0.

So suppose that T(x) exists and let us define
t = lim x->infinity T(x). We then discover that in
order for f to be continuous, we must have:

f(b,t) = b^(1/b), b <= e
eta, b >= e.

And now we can see why it is difficult to prevent
T from depending on b. We want T (and therefore t)
to be independent of b, but we see that f has to
jump from f(b,t) = eta to f(b,1) = b, no matter
how large b happens to be! This is rather
difficult to maintain and yet still have f and its
derivatives to be increasing.


Yes. Experiments with towers that seem to "extract
tetraroots" of b in various tetration extensions
I've seen seem to suggest it would depend on b.
That is, a tower that gives, say tetsqrt(b) for
one b may not for another b. This is why I drop this
property in my new list.

And this brings up mike3's problem 4:

4. f(b, x) increases strictly monotonically with respect to x, as do
all it's derivatives with respect to x.

Now mike3 points out that there's a problem with
the base b = 1.5. We notice that this is slightly
greater than eta. Indeed, the problem becomes
even more evident for base b = eta+epsilon, for
very small epsilon. As mike3 has discovered,
the second partial f_xx(b,x) appears to be
negative at first, as f(b,x) increases very
slowly -- then suddenly f_xx becomes positive
and f increases very quickly, as we would expect
of a tetration function.

Now I conjecture that the point of inflection
of y(x) = b^^x occurs exactly at y = e -- that is,
tetration increases slowly as long as the value
is less than e, and then once it crosses e it
increases quickly.



So then I came up with this alternate list of properties:
---
For real numbers x >= 0 and b > 1, a real-valued function f(b, x)
is called the "tetrational function to the base b" if it satisfies
(references to "all" x or "all" b and "everywhere" refer only to the
ranges of b and x specified):
1. Has base b, that is, f(b, 1) = b.
2. Corresponds to iterated exponentiation: f(b, x + 1) = b^f(b, x),
and f(b, x - 1) = log_b(f(b, x)).
3. Is perfectly smooth in both x and b (infinitely differentiable
everywhere.)
4. For all x, df^2/dx^2 is either >= 0 everywhere, <= 0 everywhere,
or both, but if both must have exactly one point where it changes
from one to the other.
---
What do you think? Does an f(b, x) satisfying all the above exist?
If so, is it unique? Why?

My conjecture does adhere to mike3's new property 4,
namely that y(x) = f(b,x) has at most one point of
inflection, namely at y = e, if b > eta. Of course,
if b <= eta, then y(x) has no point of inflection.

Yes. So now the question is, is the function that satisfies
1-4 above _unique_? It appears that functions satisfying
1-3 are not unique, so the clincher is 4. Is it strong enough
to make a unique real-valued function of reals b > 1 and x >= 0?

.

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