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From: monta@cmu-cs-g.ARPA (Peter Monta)
Newsgroups: net.math
Subject: Square root of exp?
Message-ID: <225@cmu-cs-g.ARPA>
Date: Sat, 26-Jan-85 19:36:20 EST
Article-I.D.: cmu-cs-g.225
Posted: Sat Jan 26 19:36:20 1985
Date-Received: Wed, 30-Jan-85 06:34:53 EST
Organization: Carnegie-Mellon University, CS/RI
Lines: 15

A nice application of spectral theory is the definition of continuous
functions of linear operators like the square root and the exponential.  I
got to wondering whether anything could be said about nonlinear operators.

Can you find, for instance, a square root of the exponential on the reals?
That is, find a function f from the reals to the reals such that
           x
f(f(x)) = e  for all real numbers x?  I sketched for a while on graph paper,
and it seems to me that something like this really should exist: limit at
minus infinity of -1, f(-1)=0, f(0)=1/e, f strictly increasing.  Failing
an explicit answer, is there a way to compute f?

Peter Monta
ARPA: monta@cmu-cs-g
UUCP: ...!rochester!cmu-cs-pt!cmu-cs-g!monta