+- Tetration Forum (https://tetrationforum.org)
+-- Forum: Tetration and Related Topics (https://tetrationforum.org/forumdisplay.php?fid=1)
+--- Forum: Mathematical and General Discussion (https://tetrationforum.org/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