 exp^[1/2](x) uniqueness from 2sinh ? - 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: exp^[1/2](x) uniqueness from 2sinh ? (/showthread.php?tid=879) exp^[1/2](x) uniqueness from 2sinh ? - tommy1729 - 06/03/2014 A possible uniqueness critertion for exp^[1/2](x) ? For x > 1 and any integer n >= 0 : 1) e/n! > d^n exp^[1/2](x)/d^n x @ x = 1 > 0. 2) 2sinh^[1/2](x) + d 2sinh^[1/2](x)/dx - exp(-x) > exp^[1/2](x) > 2sinh^[1/2](x). ( 2sinh^[1/2](x) is computed with the koenigs function ) 3) exp^[1/2](z) is holomorphic for Re(z) > 1/2. If the uniqueness fails the question is if the conditions are too strong or too weak. And if it can be improved. regards tommy1729 RE: exp^[1/2](x) uniqueness from 2sinh ? - tommy1729 - 06/03/2014 Hmm The conditions must fail because they imply that exp^[1/2](x) is entire which it is not. Not sure how to bound the derivatives then ... reduce condition 1) to d^n exp^[1/2](x)/d^n x @ x = 1 > 0 ? regards tommy1729