06/15/2014, 06:31 PM

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

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