07/25/2014, 02:46 PM
(This post was last modified: 08/02/2014, 02:48 PM by sheldonison.)

(07/24/2014, 10:47 PM)tommy1729 Wrote: The conclusion is that if | f(f(z)) - exp(z) | << 1 where sheldon's picture claims it to be , then there are no nonreal zero's for f(z).

Also the converse : If there are no nonreal zero's for f(z) then

| f(f(z)) - exp(z) | << 1 where sheldon's picture claims it to be.

I assume this was already known by sheldon but he did not post the proof.

I assumed him to be correct but double checked it. Now its certain.

I like the approach. It might be an elegent simplification of the last step of my proof. The boundary of convergence of |f(f(z))/exp(z)| would be where . Here, the amplitude of , but it is smaller everywhere else. Here is a sample calculation, where f(z) is halfway between the 2nd and 3rd zeros. Here, . If f(z) is one of the zeros, then f(f(z))=0 by definition, and .

For my proof, I am focused on the error terms for the assymptotic approximation, and a proof that all of the Taylor series terms (and Laurent series terms) converge. I have a mostly complete version of of the proofs, but its a bit involved; requires some pictures and quite a few equations. Also, there are two different dueling error terms, both of which get arbitrarily small as |z| gets larger. One of the error terms is the 1/x part of the Laurent series for the entire half iterate, which is approximately 0.003898/z-0.00172/z^2 and the other is the number of terms required in the approximation. Then I got side tracked into investigating this Laurent series, which I calculated accurate to 32 decimal digits for terms>=-20. The terms>=0 for the Laurent series are exactly the same as the entire half iterate approximation. All of the terms of the Laurent series converge. But the infinite Laurent series itself probably does not converge anywhere.

Anyway, except for all of the pictures required for the proofs, that is where I'm at. Would you be interested in the partial details so far? The last step would be to show that the approximation error term are too small to allow zeros anywhere else other than the real axis for,

- Sheldon