 Semi-exp and the geometric derivative. A criterion. - 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: Semi-exp and the geometric derivative. A criterion. (/showthread.php?tid=1181) Semi-exp and the geometric derivative. A criterion. - tommy1729 - 09/19/2017 The so-called geometric derivative from " non-newtonian calculus " : basically just f*(x) = exp( ln ' (f(x)) ) = exp( f ' (x) /f(x) ) plays a role for a criterion for the half-iterate of exp ( semi-exp ). Notice that (b^x)* = b. So in a sense it gives the average base locally. Since semi-exp grows slower than any b^x for b>1 , it makes sense that the base Goes down towards 1 as x grows. And this decrease should be smooth. So the semi-exp should satisfy For x > 1 and n a nonnegative integer. S ' (x) > 1 S " (x) > 0 x < S(x) < exp(x) Sign ( D^n S*(x) ) = (-1)^n These 4 conditions may reduce to 3 or 2, I have not looked into it. Notice I did not include analytic ! Also I did not say this gives uniqueness. But I think these conditions are intresting. And I think They have a solution. I particular I conjecture that the semi-exp computed with my 2sinh method satisfies all 4 conditions !! I tried to Find a proof but with no succes. I mentioned all this before ( apart from the conjecture about my 2sinh method ), and I might not be the first, but Im bringing this back in the spotlight. Your ideas ? Regards Tommy1729