Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.2 9/5/84; site ur-cvsvax.UUCP
Path: utzoo!watmath!clyde!burl!ulysses!unc!mcnc!decvax!genrad!panda!talcott!harvard!seismo!rochester!ur-cvsvax!bill
From: bill@ur-cvsvax.UUCP (Bill Vaughn)
Newsgroups: net.math
Subject: Re: Square root of exp?
Message-ID: <164@ur-cvsvax.UUCP>
Date: Tue, 29-Jan-85 12:39:37 EST
Article-I.D.: ur-cvsva.164
Posted: Tue Jan 29 12:39:37 1985
Date-Received: Sat, 2-Feb-85 13:12:03 EST
References: <225@cmu-cs-g.ARPA>
Organization: Center for Visual Science, U. of Rochester
Lines: 34

> 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

Z.A. Melzak's book 'Bypasses' (Wiley,1983) contains some interesting ideas
concerning this problem (see pp.89-90).  The idea is to express the
exponential function as a conjugate of a simpler function like
multipilcation.  Unfortunately he doesn't give an explict solution and the
problem of uniqueness is up in the air, but at least it's clear that
something like the animal you suggest can be defined i.e. if
exp(x) = fbf^-1(x)   where f is a function and b a constant then the ath
iterate of exp can be expressed as fb^af^-1 where now it makes sense to
talk of any real a. Therefore the 1/2 iterate of exp is  fb^1/2f^-1.

The function f has complex coefficients in its power expansion and the
value b is 0.318150 + 1.337236i (this is the smallest magnitude root of the
equation exp(x)=x).  The rest of the derivation is sketchy and most of it
left as an exercise which I haven't the time or inclination to do.

By the way, I highly recommend this book. It's got an interesting
mathematical/philosphical bent to it. His point is that the concept of
conjugacy is a very deep one and occurs in many fields.  It is also
a technique which helps makes intractable problems tractable. See also
'Companion to Concrete Mathematics', Z.A. Melzak, Wiley, 1983, pp. 51-80.