11/14/2008, 01:11 PM

bo198214 Wrote:Hi Dmitrii, though I didnt completely follow your proof it triggered coming up with a much more simplified proof. As a prerequisite we only need Picard's big theorem:Henryk, I like your proof. Indeed, it is shorter.

Picard's big theorem applied to entire functions. Each entire non-polynomial function takes on every complex number with at most one exception infinitely often.

First it can be shown that (as an entire function) does not even omit one value from . (I give the proof in a next post, though this assertion is not really necessary for the following conclusions.)

We know that for each , but by the above theorem for each there have to be infinitely other with .

Hence .

That means if we have a function , where is a superexponential with singularities only at , then has singularities outside .

Now, please, prove that some of singularities are in the right hand side of the complex plane.

Small hint:

From the asymptotic behavior, at 1<b<exp(1/e), tetration is periodic and the period is imaginary. This means, that J(z)=F^(-1)(G(z)), is also periodic.

Can it be, that h(z) is periodic (with period unity) and j(z)=h(z)+z is also periodic (with imaginary period)?

Is it possible to unwrap the periodic function with some arcsin in such a way, that the resulting function is not periodic but still entire?