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Further observations on fractional calc solution to tetration
#14
Well I dont know if you are wrong , Im just suggesting it might not be correct.

Im waiting for other experts to join the discussion. I merely expressed my doubts with arguments in the last few posts.

I dont know what Trappmann said about it and if for instance Trappmann , mike3 and sheldon support your claim that makes quite a strong case.

Im just in the state of " mathematical and general discussion ".

Im under the impression you use stronger versions of some theorems like carlson etc.
Im no expert but I have doubts on such things , in particular combined with uniqueness conditions.

For instance , I believe you said a few times that a function is determined completely by the values at integers if |f(z)| is bounded by an exponential.

That looks like a stronger version of carlson.

Again Im no expert in complex analysis , but another thing is Ramanujan's master theorem.

I have Always been told to use it with great caution.
This is mainly because the sequence f(1) , f(2) , f(3) , ... does not easily define f(-1) uniquely. For instance there many function that interpolate the factorials , but there is only one gamma function.
Then again , you might know more about this master theorem than I do. But the feeling remains with these theorems , IN PARTICULAR because not all the necc conditions are proven yet.

Maybe You use a theorem that generalizes both Carlson and Ramanujan ?

However , and forgive me if this is identical to one of your ideas , but if in the master theorem the Taylor coefficients are an interpolation of exp^[1/2](n) that is entire and has all derivatives at 0 > 0...

Then carlson's theorem applies and we have uniqueness ... therefore we have from the master theorem the values of the fake ( but correct at positive integers) exp^[1/2](-n).

At least thats what I think.

That way It relates to recent ideas of myself and sheldon and the development of " fake function theory ".

But I dont want to force you to think in that way.
You seem to want the correct tet(z),slog(z) etc with its singularities and stuff.

However its also those singularities and stuff that prevents you from being able to use FORMALLY many theorems about analytic functions or entire functions ... So that seems to make things hard.

NOW if exp^[M](b_i) is NOT chaotic , for instance cyclic that might be controllable.

But statistically speaking its chaotic , so the measure of your potential solutions seems 0 in the complex plane.

But that is no disproof.
My understanding of slog is not complete.

I believe that in the master theorem with f(x) the function considered for Taylor development and a(z) its Taylor coefficients ( a(n) ) then

f(z) is analytic for Re(z) >= 0 and a(z) is entire and unique by carlson then the master theorem ALWAYS applies.

SO i assume these conditions to be sufficient.
But I wonder if they can be relaxed.
Maybe f(z) only needs to be analytic in a strip near the real line.
Or maybe a(z) only needs to be analytic in a half-plane containing the real line.

I guess that is in the books.

This also might relate to the continuum sums ... but I think I have said enough.


Just want to add that it makes me wonder how the entire function f(z) given by the Taylor series below looks like.

f(z) = SUM fake exp^[1/2](n)/n! z^n


regards

tommy1729
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Messages In This Thread
RE: Further observations on fractional calc solution to tetration - by tommy1729 - 06/05/2014, 08:54 PM

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