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simple equation for R -> R - tommy1729 - 08/11/2010
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 RE: simple equation for R -> R - tommy1729 - 08/11/2010
ok , i must say i did not mention my assumption that the ' theta wave ' ( kneser curve ) is "smooth looking". the shape of the theta wave is important ... i assumed for all tetration solutions not yet mapping R -> R , that the theta wave is coo apart from the numbers [0,1,b,b^b,...] and looking " gauss-like ". with gauss-like i mean that it looks like the error function , one single max value in the middle , symmetric , no local minimum. for such a shape simple methods may be usefull. such as newton iterations. at least for numerical solutions. on the other hand we are not just looking for 'a' tetration , but 'the' tetration. regards tommy1729 RE: simple equation for R -> R - sheldonison - 08/13/2010
(08/11/2010, 04:10 PM)tommy1729 Wrote: .... finding the parametric real and im part of ' kneser ' might be as complicated or even equivalent to riemann mapping.I freely admit that my higher mathematics education has been stretched way beyond its limits. My wife thinks I should go back to school, so I can get an advanced degree and teach at a University. Anyway, all that aside. I don't have a really good background in understanding the Riemann mapping theorem, this despite having stumbled upon an algorithm to generate the Riemann mapping for tetration. I gather from reading on the internet, that the constructive Riemann mapping approaches are matrix based, and very slowly converging in the case of a singularity (which of course, is our case). I would assume that except at the singularity, one could continue the function, and that the theta(z) at the real axis would have to be analytic, coo. By the way, thanks for your comments Tommy! Quote:....I'm not sure if by Kneser curve you mean sexp(z), which does have one inflection point somewhere near -0.5. If you're talking about the theta(z) function, it has a really really bad singularity. As z superexponentially approaches the singularity at integer values, the behavior gets worst and worst, than the plots I posted along with the equations. The imag(z) oscillates, going up and down, and the real(z), which looks like it might converge to a value, instead super-logarithmically oscillates, growing to +real infinity, while oscillating in the imag(z)! I'll plot an updated post some time, along with some comments. - Sheldon |