02/20/2013, 10:37 PM

It came to my mind to replace 2sinh(b x) with another similar realanalytic function F and then perform the same iterations as with the sinh method. The problem is the fixpoints. If I use 2sinh(2 x) to construct a superfunction for exp(2x) then 2sinh(2 L) =/= L. But we have the advantage that 2sinh(2 x) has only one fixpoint so entire iterations for 2sinh(2 x) exists. If however we have F(L)=L then that might make things easier ( analytic continuation ). But there is fear for other fixpoints such as 0 and/or L* resulting in a nonentire and nonunique superfunction of L(x). However if we consider a real superfunction of F(x) and we only have the fixpoints L and L* and both are repelling then it might be intresting to investigate this. Notice here that L and L* means the 2 conjugate fixpoints of the function e^bx. Note however that analytic continuation will still be neccessary.