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 simple equation for R -> R tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 08/11/2010, 04:10 PM there is a simple equation to find a real to real tetration. ( [0,oo] to [0,oo] , no negative reals of course ) it works for all solutions. lets call the solution 'kneser'. and set kneser(0) = 0 , sexp(0) = 0. real(z) = x im(kneser(z)) = 0 sexp(x) = kneser(z) thats it ! however finding the solution is something else. i assume that by riemann mapping theorem that 'this sexp' is coo if ' this kneser ' is. finding the parametric real and im part of ' kneser ' might be as complicated or even equivalent to riemann mapping. and i assume the coo prop of the parametric real and im part is equivalent to the coo prop of ' kneser ' or ' sexp '. looking at it this way , its trivial to see that riemann mapping , fourier series and the 'wave' are strongly related , not to say equivalent. the question is if the equations set kneser(0) = 0 , sexp(0) = 0. real(z) = x im(kneser(z)) = 0 sexp(x) = kneser(z) lead to new insights or alternative methods. i propose that the curve of the 'knesers' ( the parametric wave of the real outputs of all known non R->R solutions ) get a special function name or special command name ( for general iterations ). ( command could be 'knesercurve(f(x))' and in the general case its very likely that the curve cannot be computed if the superfunction cant , but who knows ! ) i wanted to note that the 'wave' does have some limitations and does not have total freedom as sometimes believed : it must be an analytic wave and it must be a bounded wave that converges on C ( luckily for four series ! and thus not double periodic or finite radius ) but thats not all. since we have a smooth curve the equations presented might be solvable without riemann mapping ! since we have a smooth curve we might use newton iteration. im convinced that an iterative system exists to solve such a problem that is hardly more complicated than computing a schroeder function. that is however a numerical method not suited for closed form ( unless the algoritm itself ( limit ) but prob no sum or integral ) or properties or proofs , unless by the squeezing theorem and comparing different tetration solutions... i guess we are pretty close to reunderstanding knesers solution and being able to compute it ( rather than just the non-constructive existance ). i know im not the only one doing research on it ( e.g. sheldonison ) , but i didnt want this to be left behind. and maybe i helped some people who didnt understand kneser or sheldon and do understand now. but im not finished... about that wave again because of kneser(z+1) = exp(kneser(z)) it is clear that the kneser curve has period 1. kneser(0) = 0 and kneser(1) = 1. if we draw a straith line trough them , then i assume the curve must be above or below the line and never cross it. is that true ? regards tommy1729 « Next Oldest | Next Newest »

 Messages In This Thread simple equation for R -> R - by tommy1729 - 08/11/2010, 04:10 PM RE: simple equation for R -> R - by tommy1729 - 08/11/2010, 07:03 PM RE: simple equation for R -> R - by sheldonison - 08/13/2010, 12:28 AM

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