(05/16/2014, 07:27 AM)sheldonison Wrote: [ -> ]The next graph is f(z), the asymptotic half iterate itself, using the same grid coordinates. You can see zeros for f on the real axis, as black dots, at -0.71, -4.26, and -15.21. The pattern goes on forever, as f grows at the negative real axis, oscillating between positive and negative.

From seeing the plot I was reminded of 3 ideas I had.

Basicly I find the interpretation in terms of polar cordinates often more intresting.

This leads to 3 ideas and a remark.

remark : you say " oscillating between positive and negative ".

Now I know that if a closed jordan curve path has all arguments n times then we have n zero's within that closed path.

So that suggests that all real roots have multiplicity 1 and you are in the possesion of a proof ?

Also you have claimed that there are no zero's off the real line ?

Does that have a proof ?

I think it should be provable for Re(z) > 0.

As for these 3 ideas , I consider f(z) somewhat as between z and exp(z) but also as in between a line and a circle.

Let me explain : if we consider id(z) = z then abs id(z) has contour circles of absolute value |z|.

exp(z) has vertical contour lines of absolute value.

The idea is that for Re(z) >> 0 , f(z) slightly bends those circles towards the lines and never crosses the lines by doing so.

Also the lines never intersect , just as they dont for id(z) and exp(z).

So we have for a >> 0 , |f(a)| =< |f(a+bi)|.

That also implies there are no zero's for f when a >> 0 !!

So far the absolute value and the zero's of f.

What else seems logical ?

also for a >> 0 ;

idea 2 : While b increases arg(f(a+bi)) takes on all values infinitely often.

The number of times arg(f(a+bi)) takes on a particular angle (for fixed a) is of the form O ( x0 + x1 b + x2 f(x3 b) + x4 exp(x5 b) ) where the x_i are constants.

This idea also comes from considering the bending of the circles to the lines and from the periodicity of exp. And also from the winding , because f has an infinite amount of zero's in the left plane.

In particular the integral f ' (z)/f(z) over the path from - oo i to + oo i is intresting.

Unfortunately this is not a zeta function nor general dirichlet series ?

Speaking of dirichlet series :

idea 3 : from the previous ideas it seems f(z) cannot be of the form

ln(a0 + a1 (c1)^z + a2 (c2)^z + ...)

for Re(z) >> 0 and all a_i , ln(c_i) > 0.

If however the ideas about absolute value are wrong it is possible !

Those are the ideas I had.

It seems some theorems from complex analysis and root finding algoritms might be needed.

Although as Always the functions considered are nontrivial , nonelementary etc. but general theorems might work.

Now for the truncated Taylor series ( polynomials ) we get that abs has a simple structure for large z ; we approach a circle.

Same for the truncated hadamard , since that is also a polynomial.

So , should we study polynomials and or small z together with the absolute value ?

Or is that the wrong way ?

Another issue is that polynomials have zero's while we might want to consider f where it is not 0. that might affect the quality of the approximation of the abs.

Final remark :

Isnt there some theorem that says all zero's are have negative real part for polynomials with positive coefficients or such ?

I seem to recall something like that.

regards

tommy1729