f(f(x)) = e^x

[Fred Mellender]

How does one go about finding the function f, given f(f(x)) = e^x ?
Some sort of series expansion for f is desired. f is a function of a real
variable.

Is there a general attack on the problem: f(f(x)) = g(x), with g given?

[Max137]

Fred Mellender <no_spa...@frontiernet.net> wrote in message
news:828n9u$qvg$1...@node17.cwnet.frontiernet.net...

One method that sometimes works is to convert to a difference equation.

Define u(n)=x, u(n+1)=f(x) ......(A)

Then f(f(x))=u(n+2)=g(u(n))

If you can solve that to give u(n) as a function of n, you can then
eliminate n from equations (a) above. You sometimes have to be bold (eg
taking logs to base i) but it does work

[G. A. Edgar]

From an old post by Zdislav V. Kovarik:

Marek Kuczma: Functional Equations in a Single Variable
Monografie Matematyczne 46, Warsaw 1968
In Ch. XV, Sec. 6, he writes [modified for ASCII format]:
"For the equation
(*) f^2(x) = e^x,
a real analytic solution has been found by H. Kneser.
This solution, however, is not single-valued (Baker)
and, as pointed out by G. Szekeres, there is no
uniqueness attached to the solution. It seems reasonable
to admit f(x)=F^(1/2)(x), where F^u is the regular
iteration group of g(x)=e^x, as the "best" solution of
the equation (*) (best behaved at infinity). However,
we do not know whether this solution is analytic for
x>0.

[Kuczma defines and discusses regular iterations at
infinity in Chapter IX, Sec 5.]

References:

Baker, I.N.: The iteration of entire transcendental
functions and the solution of the functional equation
f(f(z))=F(z), Math. Ann. 120(1955), pp. 174-180

Kneser, H.: Reele analytische Loesungen der Gleichung
f(f(x))=e^x und verwandten Funktionalgleichungen,
J. reine angew. Math. 187(1950), pp. 56-67

Szekeres, G.: Fractional iterations of exponentially
growing functions, J. Australian Math. Soc.
2(1961/62), pp. 301-320

--
Gerald A. Edgar ed...@math.ohio-state.edu

[Mike Morris]

On Fri, 3 Dec 1999 "Fred Mellender" wrote:
>How does one go about finding the function f, given f(f(x)) = e^x ?

>Some sort of series expansion for f is desired?

It sounds like your teacher has been browsing at
http://www.mathpages.com/home/kmath507.htm

[David Petry]

Fred Mellender wrote:

> How does one go about finding the function f, given f(f(x)) = e^x ?

Here's an approach to the problem that does seem to yield a "best"
solution.

First, find a solution to g(g(x)) = exp(x) - 1. We can find a formal
power series for g by setting g(x) = x + c2 x^2 + c3 x^3 ... and
plugging that in to the equation, and solving for each ck.

Then, if the power series so obtained has a positive radius of
convergence, the function can be extended to all positive reals
by the equation g(exp(x)-1) = exp(g(x)) -1.

Since exp(x) -1 grows at the same rate as exp(x), we might
expect g(x) to grow at the same rate as the function f(x) that we
are seeking, which suggests that we define f(x) as

f(x) = (lim n->oo) log^n( g ( exp^n (x) ) )

where exp^n(x) = exp( exp( ... exp(x) ...)) (n compositions)

So now we need to determine if the power series for g converges.

Disclaimer: 75% of what I post in sci.math is crap :-(