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 computing f(f(x)) = exp(x) with g(g(x)) = - exp(x) tommy1729 Ultimate Fellow     Posts: 1,370 Threads: 335 Joined: Feb 2009 06/03/2009, 09:36 PM tommy1729 is back with a new idea as the title said : computing f(f(x)) = exp(x) with g(g(x)) = - exp(x) or at least trying , my idea is in " beta phase ". so why should or could g(x) and f(x) be related you probably wonder. well , as said , im not sure yet , but i will try to clarify how i got the idea. first , the superfunction of exp(x) and - exp(x) relate. to be more precise ; like this : slog ( x + pi i ) = slog ( exp(x + pi i) ) - 1 = slog ( - exp (x) ) - 1 " slog ( - exp (x) ) - 1 " seems the superfunction of - exp(x). second , - exp(x) has a real fixpoint ! thus we can use regular iteration ( or other fixpoint based methods ) third : slog(x + 2pi i) = slog(x) , thus slog(x) is periodic , we say has period 2 pi i. ( if defined at that point ! ) proof : slog(z + 2pi i) = slog(exp( z + 2 pi i )) - 1 = slog(exp(z)) - 1 = slog(z) fourth : slog(x) is strictly rising on the positive reals. combining the above , we might be able to compute f(x) such that f(f(x)) = exp(x) or slog(x). and maybe also arrive at the last uniqueness conditions , in the sence of a single unique slog(z) as the general solution for tetration slog(z) + C ( C some constant ) thats the main idea , i dont expect the function to be entire however analytic continuation might be possible , although that may loose the conditions above on the domain where the continuation takes place ( but who cares ). if analytic continuation is not possible than probably a real analytic one is still possible ... i think ... regards tommy1729 tommy1729 Ultimate Fellow     Posts: 1,370 Threads: 335 Joined: Feb 2009 06/06/2009, 12:28 PM i would like to note that slog cannot be entire nor meromorphic on C AND (!) denote complex iterations consistantly at the same time. the reason is once again the period , if slog is meromorphic on C then we can solve for slog(x) = 1 , and also slog(x2) = 1 + 2pi i and slog(x_k) = 1 + 2pi i k in general. but exp( 1 + 2pi i k) = e for all k. let slog(Q1) = Q2 , then complex iterations requires that for any complex Z1 : if slog(Q1) = slog(Q1 + Z1) = Q2 , then slog(Q1 + k Z1) = Q2 assume 3 distinct Z1 Z2 Z3 and integer k1 k2 k3 : Q2= slog(Q1) = slog(Q1 + Z1) = slog(Q1 + Z2) = slog(Q1 + Z3) which implies : Q2= slog(Q1 + k1 Z1) = slog(Q1 + k2 Z2) = slog(Q1 + k3 Z3 ) ( also 2pi i periodic which may or may not be a Z1 Z2 Z3 value , doesnt matter ) but that is impossible !!! since that would be a 3-periodic meromorphic function on C ! thus slog if meromorphic , must be bounded by vector additions and inequalities to avoid 3-periodic behaviour and yet be 2pi i periodic. ( since log(0) = oo entire is totally ruled out ) it might very well be that this tread has an ideal base for tetration ; for which my comments are more usefull than for other bases. ( fixed point within period or not comes to mind ) further example assume slog is only defined for -2 pi i < im(x) < + 2 pi i but if slog(A_k) is within this zone , and slog(A_k) = 1 + 2pi i k is solvable for all k , then this is a kind of paradox. since then A_k has to be dense in a mainly vertical way. which loses the property of Coo ( unless lineair ) at that set !!! in fact , if A_k are ' close ' to eachother , it tends to be constant in that zone. so we end up with - i guess - slog(z) is defined for -2 pi i < im(z) < + 2 pi i and maps only to -2 pi i < im(slog(z)) < + 2 pi i regards tommy1729 tommy1729 Ultimate Fellow     Posts: 1,370 Threads: 335 Joined: Feb 2009 06/06/2009, 10:28 PM (06/06/2009, 12:28 PM)tommy1729 Wrote: i would like to note that slog cannot be entire nor meromorphic on C AND (!) denote complex iterations consistantly at the same time. the reason is once again the period , if slog is meromorphic on C then we can solve for slog(x) = 1 , and also slog(x2) = 1 + 2pi i and slog(x_k) = 1 + 2pi i k in general. but exp( 1 + 2pi i k) = e for all k. let slog(Q1) = Q2 , then complex iterations requires that for any complex Z1 : if slog(Q1) = slog(Q1 + Z1) = Q2 , then slog(Q1 + k Z1) = Q2 assume 3 distinct Z1 Z2 Z3 and integer k1 k2 k3 : Q2= slog(Q1) = slog(Q1 + Z1) = slog(Q1 + Z2) = slog(Q1 + Z3) which implies : Q2= slog(Q1 + k1 Z1) = slog(Q1 + k2 Z2) = slog(Q1 + k3 Z3 ) ( also 2pi i periodic which may or may not be a Z1 Z2 Z3 value , doesnt matter ) but that is impossible !!! since that would be a 3-periodic meromorphic function on C ! thus slog if meromorphic , must be bounded by vector additions and inequalities to avoid 3-periodic behaviour and yet be 2pi i periodic. ( since log(0) = oo entire is totally ruled out ) it might very well be that this tread has an ideal base for tetration ; for which my comments are more usefull than for other bases. ( fixed point within period or not comes to mind ) further example assume slog is only defined for -2 pi i < im(x) < + 2 pi i but if slog(A_k) is within this zone , and slog(A_k) = 1 + 2pi i k is solvable for all k , then this is a kind of paradox. since then A_k has to be dense in a mainly vertical way. which loses the property of Coo ( unless lineair ) at that set !!! in fact , if A_k are ' close ' to eachother , it tends to be constant in that zone. so we end up with - i guess - slog(z) is defined for -2 pi i < im(z) < + 2 pi i and maps only to -2 pi i < im(slog(z)) < + 2 pi i regards tommy1729 seems i wrote slog(z) , i meant inv slog(z) sorry. not used to this 'new function' yet riemann surfaces should be handy i think ... still thinking ... tommy1729 « Next Oldest | Next Newest »

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