Newsgroups: sci.math From: pmontgom@cwi.nl (Peter L. Montgomery) Subject: Re: Weird Functions Date: Sat, 2 May 1998 08:03:25 GMT In article <01bd758c$349fc8c0$1825a3ce@familycomp> "Matt Denham" writes: >I assume you know of functions that are of the form x=f(f(f(x))), or: >The most elementary example of this is: > >f(x)=x-1/x > >where x is not 0 or 1 (these degenerate to part of their own triad, which >also includes -1/0). > >Can somebody give a function similar to this, but x=f(f(f(f(f(x))))) >[pentads of numbers that are cyclic by this function]? > Recall tan(x) + tan(y) tan(x + y) = ---------------- 1 - tan(x) tan(y) whenever tan(x), tan(y), tan(x + y) are all defined. Temporarily leaving t fixed but unspecified, try x + tan(t) f(x) = ------------- 1 - x tan(t) Then f(tan(x)) = tan(x + t) whenever everything is defined, This holds for all x, so (f^5)(tan(x)) = tan(x + 5*t) where f^5 denotes the five-fold composition of f. The tangent function has period pi, so let t be a multiple of pi/5. That is, tan(t) = +- sqrt(5 +- 2*sqrt(5)). -- Peter-Lawrence.Montgomery@cwi.nl San Rafael, California ============================================================================== From: fredh@ix.netcom.com (Fred W. Helenius) Newsgroups: sci.math Subject: Re: Weird Functions Date: Sat, 02 May 1998 10:08:17 GMT "Matt Denham" wrote: >I assume you know of functions that are of the form x=f(f(f(x))), >The most elementary example of this is: >f(x)=x-1/x You mean (x-1)/x, of course. >where x is not 0 or 1 (these degenerate to part of their own triad, which >also includes -1/0). > >Can somebody give a function similar to this, but x=f(f(f(f(f(x))))) Very similar indeed: let phi = (sqrt(5) + 1)/2, then f(x) = (phi*x - 1)/x will do the trick (except for x=0,1,phi,1/phi). Substituting 1/phi, -phi or -1/phi for phi will also work. -- Fred W. Helenius