Some "Theorem" on the generalized superfunction

[Leo.W]

Hey, Leo, very interesting stuff. Don't worry about your English, you speak very well; it's alright.

To add in Latex on this forum just write

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As to what you do with \( \zeta(\tan(z)|\exp(\exp(z))) \); that's very clever. That's particular to what MphLee and I were talking about. I guess my question was more so, how do you plan to choose which superfunction we produce? That was more what I was asking. 

I get that,

\(
\tan(F(z)) = F(z+1)\\
\exp(\exp(G(z))) = G(z+1)\\
\zeta(\tan(z)|\exp(\exp(z))) = F(G^{-1}(z))\\

\)

I was wondering if you had any method where it chooses a particular super function \( F \) and \( G \).

Regards, James

Thanks a lot, James! 


(Maybe here's about how I choose a superfunction, I don't know if I expressed it in a clear way?)
I suggest we denote \( \alpha\{f(z)\}(z) \) as the specific abel function of f(z), and \( \sigma\{f(z)\}(z) \) as the schroder function and so on.
Straightforwardly, the superfunction \( G \) is exactly (probably) equal to \( \alpha^{-1}\{\exp(z)\}(\frac{z}{2}) \).

Choosing the appropriate branch cut of  \( \alpha^{-1}\{tan(z)\}(z) \) is kind of tricky. First of all, we should choose a branch cut which has the largest range, so an entire cut that contains a whole complex plane is better than a cut is either not entire or its range is only a subset of \( \mathbb{C} \), the advantage of this method is that the \( \zeta\{\tan(z)|\exp(\exp(z))\} \) we want would be entire, otherwise it may contain branch cuts or undefined for some values.

For instance, the superfunction of sine function has infinitely many branch cuts, the most common or most times generated one is that \( T(z)=\alpha^{-1}\{sin(z)\}(z)=\sqrt{\frac{3}{z}}+\mathrm{o}(\frac{1}{z}) \). By iterating \( T(z)=\sin^{-1}(T(z+1)) \) we have this branch cut. However it is not entire, in fact the range of this function contains no negetive real numbers. The range is about to stay in the area approximately \( \arg(z)\in[-\frac{\pi}{3},\frac{\pi}{3}] \).
Another branch cut which is pretty approachable is with the initial condition \( T(0)=i \), which is connected to the superfunction of sinh(z): \( \alpha^{-1}\{sin(z)\}(z)=i\alpha^{-1}\{\sinh(z)\}(z) \), and if this cut is an entire function, we'll choose it instead of the former one.

However, for some reason we have to choose those non-entire cuts, if you want to generate a real-to-real \( \zeta \), sometimes it's inevitable to use such functions.

For a real-valued function, it's better to choose a real-to-real superfunction than a real-to-complex superfunction, e.g. the\( G^{-1}(z)=\frac{1}{2}\alpha\{\exp(z)\}(z) \), where we use the merged version of \( \alpha\{\exp(z)\}(z)=slog_e(z) \).

So the order of what kind of superfunctions I choose is,

if f is real-to-real, T:
real-to-real(entire,merged)>real-to-complex(entire,merged)>real-to-complex(entire,unmerged)
>real-to-real(not entire)>real-to-complex(not entire)

if f is real-complex, T:real-to-complex(entire, range is the whole plane,merged)>real-to-complex(entire, range is subset of C)>real-to-complex(not entire)

And for a function, its superfunction predicted to have an oscillating behavior(such as 0.1^x), real-to-complex(entire,merged) is the best choice to use.

For the merged function, we use the very 2 fixed points having the least absolute value and larger real part.
For real-to-real function, we'd better use the fixed point whose multipler s>1, since 0<s<1 could cause the function to be not entire, and s<0 causes the oscillating behavior.

The pics show the superfunction(real-to-real) of tan(z), an entire function(you can see many polynomial singularities and zeroes, there's no cuts however, The real axis behaves like a cut owing to the oscillation between sigularities and zeroes.)(produced by WolframMMA12.2 ComplexPlot)

Regards, Leo