Further observations on fractional calc solution to tetration

[mike3]

Im sorry Mike, but I think switching to slog is not going to help.
Convergeance is not the big problem here , the big problem is the selfreference.

I see no correlation between the convergeance issue and the selfreference issue.

As you adequately put :

The integral involves a complex tetration. Yet, we don't have complex tetration in the first place if we are trying to construct it -- so how does this work?

( quote slightly modified )

Ergo without a fundamental change in the strategy , I see no future for this method at the moment.

After a countable amount of efforts I gave up on trying to fix this anomaly.

Selfreference is the red wire and déjà-vu in hard complex analysis.
It requires the action of a master.

regards

tommy1729

Yes, however I believe, given this is a continuation of the previous thread, what is being done is to hypothesize that a tetration function satisfying certain criteria exists. Namely, he hypothesizes that there exists a tetration function \( F(z) \) satisfying the criteria

(0. \( F(z) \) satisfies the tetration functional equations and is holomorphic on at least a cut plane, so \( F(0) = 1 \) and \( F(z+1) = e^{F(z)} \))
1. \( |\frac{1}{F(z)}| \le Ce^{\alpha |\Im(z)|} \) for \( 0 \le \alpha < \pi/2 \), \( \Re(z) < -1 \).
2. \( \frac{1}{F(z)} \) decays uniformly as \( \Re(F(z)) > 0 \)
3. \( \Re(F(z)) > 0 \) for \( \Re(z) > -1 \).

Then he works from that hypothesis to a formula for that function using his fractional calculus methods. In particular, using his fractional calculus results he gets the first formula given on his first post in this new thread from the above hypotheses, and then works from there.

(--- Note, this is a slight derail as you were talking about circularity, but I just noticed this! ---)
Now, Kneser's function looks to show (I don't have a proof on hand) the existence of a function satisfying criteria 0, 1, and 3. The problem with this is that there does not exist a tetration function which also satisfies criterion 2! This is a consequence of the chaotic nature of the exponential map (the fact that the Julia set \( J[\exp] \) is the whole complex plane \( \mathbb{C} \) so it is chaotic everywhere.).

So his method looks to start from a flawed premise, and therefore it is no surprise it does not converge. I just realized this as I hadn't quite paid close enough attention to his criteria before to notice the criterion (2) above.

(Now, if you, JmsNxn, or anyone else can shoot down my argument above as to why a tetration function satisfying the above criteria doesn't exist, I'd be happy to hear about it. Though I don't think the Kneser function would be the one that would work, since at least from the graph its reciprocal does not appear to satisfy hypothesis 2.)

EDIT: I also notice that, as strictly worded, criterion 1 does not apply either due to the pole at \( z = -1 \), but it would work for \( \Re(z) < -1 - \epsilon \).

(--- End derail ---)

This does not look circular, since you can try to start from a series of hypotheses to attempt to construct an object satisfying them. If the object is constructed successfully, then that shows that the statement "there exists an object satisfying these hypotheses" is true. In this case, it is not, but that does not matter with regard to the validity of the underlying method in general. The reasoning is:

--
We seek the construction of an object satisfying some hypotheses.
1. Deduce from the hypotheses and known results an equation which an object satisfying them would also satisfy, and such that an object satisfying the formula would also satisfy the original hypotheses.
2. If the equation can be gotten to a form where it involves only known quantities on one side and the hypothesized object on the other, attempt to calculate a solution. If the solution can be obtained, then we have an object satisfying the given hypotheses.
--

The argument is not circular. By using truths already proven, it essentially restates the hypotheses in a different form, and then by solving that formula which is equivalent by logic to the original hypotheses. At least that's what I get from it, anyways. I'm not sure if my above description is entirely right but hopefully it should show why this is not circular.

Although, what he gave in the first post does not appear to be complete since he hasn't yet gotten the formula to a form involving only known quantities such as only the integer (discrete) values of tetration, which follow immediately from criterion 0.

(derail)

(although if the premise is flawed this is not going to go anywhere anyways -- I'm just saying)

(/derail)