How many methods have this property ? - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: How many methods have this property ? (/showthread.php?tid=872) How many methods have this property ? - tommy1729 - 05/22/2014 How many C^oo ( or analytic ) solutions to tetration satisfy : For all bases larger than eta for which the method works : Property : sexp_a(x) > sexp_b(x) For all a>b and x>1. I used convention sexp_a(1) = a , sexp_b(1) = b here. I could not find it here ? I assume this property fails for base change and kneser. I think it works for Cauchy contour method (kouznetsov?) and 2sinh method (tommy1729). Maybe this has been said long ago. regards tommy1729 RE: How many methods have this property ? - sheldonison - 05/22/2014 (05/22/2014, 08:43 AM)tommy1729 Wrote: How many C^oo ( or analytic ) solutions to tetration satisfy : For all bases larger than eta for which the method works : Property : sexp_a(x) > sexp_b(x) For all a>b and x>1. I used convention sexp_a(1) = a , sexp_b(1) = b here. I could not find it here ? I assume this property fails for base change and kneser. I think it works for Cauchy contour method (kouznetsov?) and 2sinh method (tommy1729). Maybe this has been said long ago. regards tommy1729 I think all tetration methods have your property: Quote:Property : sexp_a(x) > sexp_b(x) For all a>b and x>1. There are anomalies, but the definition is different. The typical Kneser method anomalies involve sexp(x)=sexp(y), x<>y, a>b, but counter intuitively, sexp_a(x+0.5) sexp_10(2.3637006685629+0.5) The solutions which avoid this anomaly are coo, but not analytic, like the basechange method, which derives all tetrations for all bases from a single master base. Originally, for the basechange solution we used cheta, for exp(1/e), but this is really arbitrary. Then if f is the arbitrary chosen superfunction for the arbitrarily chosen master base, we can define a family of tetrations for all bases>exp(1/e), which avoid the anomaly. For example, f(z) could be 2sinh^z, or f(z) could be sexp(z) for any base. $k(a)=\lim_{n \to \infty} f^{-1}(a\uparrow\uparrow n)-n$ $\text{sexp}_a(z) = \lim_{n \to \infty} \log_a^{on}f(z+k(a)+n)$ - Sheldon