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From: lambert@boring.UUCP
Newsgroups: net.math,net.puzzle
Subject: Re: Exponent Duals
Message-ID: <6359@boring.UUCP>
Date: Fri, 15-Mar-85 10:03:35 EST
Article-I.D.: boring.6359
Posted: Fri Mar 15 10:03:35 1985
Date-Received: Sun, 17-Mar-85 21:07:05 EST
References: <555@ahutb.UUCP>
Reply-To: lambert@boring.UUCP (Lambert Meertens)
Organization: CWI, Amsterdam
Lines: 27
Xref: linus net.math:1537 net.puzzle:505
Summary: 
Apparently-To: rnews@mcvax.LOCAL

> Define an "exponent dual" of the x to be some OTHER number y so that
> 	x^y = y^x

An amusing problem concerning exponent duals:

Define a real function f such that

   i)   for each z <> 0, the exponent dual of f(z) is f(-z);
   ii)  all exponent duals can be obtained in this way;
   iii) the expression defining f(z) is closed and analytic
        (that is, is built from standard mathematical operations
	and functions; no limits or infinite sums).

Solution (rot 13):

Gur inyhr bs gur shapgvba s nccyvrq gb m vf:

             rkc bs {m bire [(rkc bs m) zvahf bar]}.

Gur cnve (gjb, sbhe) vf bognvarq ol gnxvat

	     m = +/- 0.693147180559945
-- 

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