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From: Transfer Principle on 2 Apr 2010 21:21
On Mar 26, 12:04 pm, Marc Olschok <nob...(a)nowhere.invalid> wrote:
> William Elliot <ma...(a)rdrop.remove.com> wrote:
> > Let's exponentate functions, f^2 = f o f.
> > How far would you want to push it?
> > What's f^-1?  What's 0^1?  What's f^sqr 2? ...
> As for noninteger exponents, I do not know any sensible defnition
> for such a situation.

This is another topic that's popular with fans of zeration and
tetration, and unpopular with standard theorists.

First of all, let me use a different notation. Since the
notation f^r is often ambiguous (for example, compare sin^2
with sin^-1), many tetraters prefer to write the symbol "o" as
a subscript to emphasize that the operation is composition and
not multiplication. In ASCII this can be rendered as f_o^r, so
we shall write f_o^2 for fof, f_o^-1 for f inverse, and so on.

As Elliot has already alluded to in this thread, we can define
f_o^(1/2) (often written as sqrt(f), just as with real numbers)
to be a function g such that gog = f. But here, the standard
theorists like to point out that such a g is seldom, if ever,
unique, even if we restricted g to the continuous functions. In
fact, there are usually _infinitely_ many possible continous
functions for g. Typically, the value of g on a certain interval
determines the function on the entire real line. Because of this
non-uniqueness, most standard theorists are ready to declare the
definition of f_o^(1/2) impossible.

But, much to the standard theorists' dismay, there are methods of
determine a suitable g for certain analytic functions f -- and
these methods are perfectly _rigorous_. One method, called
regular iteration, entails finding a fixed point of f -- i.e., a
real number x such that f(x) = x -- and then finding a Taylor
series for f centered at that point.

One common example of a function f for which sqrt(f) has been
calculated is the dexp function, defined as dexp(x) = e^x-1. It
has a real fixed point x=0 and Taylor series:

f(x) = x + x^2/2 + x^3/6 + x^4/24 + x^5/120 + x^6/720 + O(x^7)

Then we can use regular iteration to find the function:

g(x) = x + x^2/4 + x^3/48 + x^5/3840 - 7x^6/92160 + O(x^7)

The following link (Sloane's integer sequences) gives more
information about this function g = sqrt(f):

http://www.research.att.com/~njas/sequences/A052105

Regular iteration has also been used to determine sqrt(f) for
f = sqrt(2)^x (fixed points 2,4) and f = e^(x/e) (fixed point e).

Another method that can be used is the matrix method. Current
sci.math poster Gottfried Helms often uses the matrix method for
his continuous iteration, which entails linearizing f and its
derivatives as a matrix, then performing various operations on
the matrix. One can perform a Google search for Gottfried Helms
and the matrix method for more info on how this method works.

Thus, it is possible to find sqrt(f) and f^r for certain real
values of x, despite the standard theorists saying otherwise.
From: elmerturnipseed on 5 Apr 2010 00:32

I'm very sorry to report that most of the replies here are way over my
head. I'm still at high school math level.

Thanks, Norbert for explaining my mistakes.

Transfer Principle, it looks like you showed the flaw in my theory,
and I thank you. Your explanation is temporarily beyond me, but now I
get to go explore zeration.

William Elliot asks: "What's the source of the article?"
I'm sorry I didn't bookmark the article, but it's by Gerard Michon. He
proved that 0^0=1 based on a rule that I thought depended on the
notation system used. I figured a different system might change the
rule.

Dr. Michon very kindly replied to my email and explained that by
changing the language of math, your communication is flawed.

I replied asking that he consider a planet that has always used
Turnipseed notation, and they consider our notation flawed. I wonder
why our two systems have different amounts of exponentials. I wonder
if these human-generated exponentials really "exist", or if we only
bother with them because our notation suggests they exist.

5^1 is a number multiplied by itself zero times. 5.

So what is a number multiplied by itself less than zero times? Should
not 5^0 be considered the first negative exponential?

Thanks very much for your replies and patience,
Cheers,
Elmer Turnipseed
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