Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.1 6/24/83 (MC840302); site boring.UUCP
Path: utzoo!watmath!clyde!burl!ulysses!mhuxr!mhuxb!mhuxn!mhuxm!mhuxj!houxm!whuxlm!harpo!decvax!genrad!panda!talcott!harvard!seismo!mcvax!boring!lambert
From: lambert@boring.UUCP
Newsgroups: net.math
Subject: Re: Square root of exp?
Message-ID: <6298@boring.UUCP>
Date: Mon, 28-Jan-85 15:06:30 EST
Article-I.D.: boring.6298
Posted: Mon Jan 28 15:06:30 1985
Date-Received: Thu, 31-Jan-85 01:31:32 EST
References: <225@cmu-cs-g.ARPA>
Reply-To: lambert@boring.UUCP (Lambert Meertens)
Organization: CWI, Amsterdam
Lines: 69
Summary: 
Apparently-To: rnews@mcvax.LOCAL

A general construction for defining a function f such that
f(f(x)) = g(x) for a given strictly increasing function g
was given in

    G.H. Hardy, Orders of Infinity, Cambridge Tracts in
    Math. and Math. Physics, No. 12, University Press,
    Cambridge, England, 1910 (2nd ed. 1924).

Start with a pair of reals x0 and x1 such that x0 < x1, and
define f on [x0,x1] by taking any strictly increasing
function satisfying

    f(x0) = x1, f(x1) = g(x0).                      (1)

(So we can start by taking the linear function on the
interval [-1, 0] such that f(-1) = 0 and f(0) = 1/e.)

Let h stand for the functional inverse of f.  The function h
is now defined on the interval [x1, g(x0)].  By using the
identity

    f(x) = g(h(x))                                  (2)

we know the values of f(x) on the interval [x1, g(x0)].  The
function h is now also defined on [g(x0), g(x1)] and the
process can be repeated indefinitely.  For determining f(x)
for values of x < x0, basically the same method is used, but
now by using, instead of (2), the identity

    f(x) = h(g(x)).                                 (3)

Since (even for fixed values x0 and x1) there are many
functions satisfying (1), the solution is by no means
unique.  However, it turns out that for the square root of
the exponential function there is an analytic solution (i.e.,
a function f such that f is analytic):

    H. Kneser, Reelle analytische Loesungen der Gleichung
    phi(phi(x)) = e^x und verwandter Funktionalgleichungen
    (Real analytic solutions of the equation phi(phi(x)) =
    e^x and related functional equations), J. Reine Angew.
    Math. 187 (1948), 56-67.

In fact, Kneser solves a more general problem: he constructs
a function PSI satisfying

    PSI(exp(z)) = PSI(z) + 1.

The function PHI defined by

    PHI(z) = PSI^-1.[PSI(z)+1/2]

is analytic and real over the full real axis, and satisfies

    PHI(PHI(z)) = exp(z).

In general, we can now define

    (exp^a)(x) = PSI^-1.[PSI(x)+a],

and we have

    (exp^1)(x) = exp(x),
    (exp^a)((exp^b)(x)) = (exp^(a+b))(x).
-- 

     Lambert Meertens
     ...!{seismo,philabs,decvax}!lambert@mcvax.UUCP
     CWI (Centre for Mathematics and Computer Science), Amsterdam