[2014] The secondary fixpoint issue. - 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: [2014] The secondary fixpoint issue. (/showthread.php?tid=887) [2014] The secondary fixpoint issue. - tommy1729 - 06/15/2014 Let exp(x) =/= x , exp(exp(x)) = x , Im(x) > 0. Thus x is a secondary fixpoint in the upper half-plane. Let y be a number in the upper half-plane such that sexp(y) = x and sexp is analytic in the upper half-plane. Then by assuming the existance of y and the validity of sexp(z+1) = exp(sexp(z)) everywhere we get : sexp(y+1) =/= sexp(y) , sexp(y+2) = sexp(y). thus for y = a + b i and R any real , we have that : sexp(R + b i) is a nonconstant analytic PERIODIC function in R. And then by analytic continuation , sexp(z) is a nonconstant analytic periodic function in the upper half-plane !! But that cannot be true !! Same applies to n-ary fixpoints !! Seems unlikely that sexp contains none of these n-ary fixpoints ?! And as for the functional equation f(x+1) = exp(f(x)) + 2pi i that is on another branch. So that does not seem to help. Where is the mistake ?? I posted this before , hoping the seeming paradox is understood now. Also note that the "paradox" is not limited to 2nd ary fixpoints , 3rd ary fixpoints etc but also V-ary fixpoints for any V > 1 ! regards tommy1729 RE: [2014] The secondary fixpoint issue. - sheldonison - 06/15/2014 (06/15/2014, 06:31 PM)tommy1729 Wrote: .... Seems unlikely that sexp contains none of these n-ary fixpoints ?! And as for the functional equation f(x+1) = exp(f(x)) + 2pi i that is on another branch. So that does not seem to help. Conjecture: if $\text{sexp}(z)=L$, than for some positive integer n, there is a non-zero integer m such that $\text{sexp}(z-n)=L+2m\pi i$ This would apply for L equals any fixed point of exp(z) and any finite value of z. This conjecture would apply to both the Kneser solution, and the secondary fixed point solution. I think it can be proven by showing for these two solutions, that sexp(z-n)<>L for large finite values of n. RE: [2014] The secondary fixpoint issue. - tommy1729 - 06/15/2014 (06/15/2014, 08:07 PM)sheldonison Wrote: (06/15/2014, 06:31 PM)tommy1729 Wrote: .... Seems unlikely that sexp contains none of these n-ary fixpoints ?! And as for the functional equation f(x+1) = exp(f(x)) + 2pi i that is on another branch. So that does not seem to help. Conjecture: if $\text{sexp}(z)=L$, than for some positive integer n, there is a non-zero integer m such that $\text{sexp}(z-n)=L+2m\pi i$ This would apply for L equals any fixed point of exp(z). This conjecture would apply to both the Kneser solution, and the secondary fixed point solution. I think it can be proven by showing for these two solutions, that sexp(z-n)<>L as n goes to infinity. Wait a minute , we have sexp(R + oo i) = L or conj(L) for all real R and all real oo !? I thought we all agreed on that for years ? But R + oo i - n is also of the form R + oo i. ?? Now Im even more confused. Thanks for the reply though. regards tommy1729