Differential equation for exp^[0.5] - 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: Differential equation for exp^[0.5] (/showthread.php?tid=903) Differential equation for exp^[0.5] - tommy1729 - 07/14/2014 We know the logarithmic derivative is f ' / f dz. We know the solve derivative is dz / ( f ' (inv f) ). Let f = exp^[0.5]. Then solve f = inv f = ln(f). THUS f ' / f = 1 / ( f ' (ln(f)) ) I wonder how good approximations of f satisfy the differential equation. (2sinh , base change , fake etc ) What about theta(z) ?? regards tommy1729 RE: Differential equation for exp^[0.5] - tommy1729 - 07/26/2014 (07/14/2014, 12:55 AM)tommy1729 Wrote: We know the logarithmic derivative is f ' / f dz. We know the solve derivative is dz / ( f ' (inv f) ). Let f = exp^[0.5]. Then solve f = inv f = ln(f). THUS f ' / f = 1 / ( f ' (ln(f)) ) I wonder how good approximations of f satisfy the differential equation. (2sinh , base change , fake etc ) What about theta(z) ?? regards tommy1729 I wonder if g ' / g = 1 / (g ' (h(g))) where h is a fake ln would make a good fake g(z) such that g^[2](z) - exp(z) is small ? regards tommy1729