05/15/2014, 09:49 PM

(05/15/2014, 06:15 PM)sheldonison Wrote: So, we have this nice asymptotic approximation, f=exp^0.5(x). Lets take the best one, version IV. What would f(f(z))/exp(z) look like???? Its an entire function, since f is entire, and since exp(z) has no zeros. The answer is really neat, and counter intuitive! I'll have to post some complex plane graphs tomorrow.

- Sheldon

I just posted why the tommy sheldon conjecture should hold.

So yes our f(x) is becoming very good.

But maybe we can learn from the classic masters something too ?

I mean surely you thought of the bell numbers right ?

Afterall if one wants boundaries on

e Bell(n)/n!

or equivalently

ln( Bell(n)/n!) + 1

Then one could use in a similar fashion

c_n x^n < exp(exp(x))

and repeat what we did here.

But the Bell numbers have been studied in different ways too , going from number theory and combinatorics to other uses of calculus and dynamics etc.

Or could the classic masters have learned from us ?

Maybe our methods give better approximations of bell numbers.

I recall that also Gottfried has studied the bell numbers in detail with matrices.

Maybe we can generalize the bell numbers in " tetration style ".

Another thing :

In NKS Wolfram mentions cellular automatons and recursions that grow at unknown growth rates.

I believe these relate to asymptotic function related to tetration and exp^[1/2] ...

Although it does not follow clearly from the equations , not even the recursive ones.

That might be an active research field.

As for f(z) and f(f(z))/exp(z) being entire ...

I repeat my questions/remarks :

What does the weierstrass product of those look like ??

Do we have f(z) = exp(g(z)) ( 1 + a_1 z)(1 + a_2 z) ... ?

or is there no exp(g(z)) term ??

Well it seems the g(z) is absurd.

HOWEVER we know g(z) could be a fake log !!

and so could ( 1 + a_1 z)(1 + a_2 z) ... be a fake 1/ln(x) !

So its not directly Obvious.

Notice f(z) = exp(g(z)) ( 1 + a_1 z)(1 + a_2 z) ...

implies f(z) has no positive real zero's but an infinite amount of negative zero's ( the a_n ).

Also f(z) = exp(g(z)) ( 1 + a_1 z)(1 + a_2 z) ... could be wrong !

The weierstrass product form could be more complicated !

Is it ?

Similar questions for f(f(z)/exp(z).

Since f(f(z))/exp(z) must be close to 1 for positive real z , it follows it must go to oo for other part of the complex plane.

( its clear from absolute convergeance that f(q z) does not grow faster than f(q) for z nonreal and q a positive real )

I bet this is what sheldon's plots will show us beautifully.

And there is also still carlson's theorem in the game

regards

tommy1729