 (draft) integral idea - 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: (draft) integral idea (/showthread.php?tid=666) (draft) integral idea - tommy1729 - 06/25/2011 this idea is a draft and perhaps somewhat farfetched ... consider a non-linear real entire function f(x) strictly rising on the reals with no more than 2 conjugate fixpoints at R + oo i and R - oo i. also f(x) is uniquely invertible on the line segment connecting the 2 conjugate fixpoints L and L*. let R be the real value of L , and x be real then f(2x) i = f((x-R)i + R) now consider the integral from R - oo i till R + oo i of f(z) dz = C_0 let C_n be the integral after substitution z = f^[1/2^n](x). now consider G = sum over +integers n : abs(C_n - C_0) G = 0 implies a uniqueness condition on the superfunction of f(x). G = 0 is equivalent to df/dx f^[1/2^n](x) >= 0 and also equivalent to df/dq f^[q](x) >= 0. how to compute or prove a sum like abs(C_n - C_0) ? i have considered taking the continuum sum of C_x - C_0 or a contour integral around it and then trying to prove the period of one of those ... this also resembles fourrier transforms but i dont know how to use transforms on this ... it would be nice if we could convert this problem to tetration , however it might be problematic ; half-iterates tend not to agree on both fixpoints , the mapping to another function might not be topologically conjugate , an integral on a riemann sphere can be closed and its equivalent on the complex plane might not ( + oo i IS - oo i at the sphere). however this is a draft and conjugate fixpoints are special cases so there might be progress possible with some " luck ". regards tommy1729