how many superfunctions? [was superfunctions of eta converge towards each other]

[sheldonison]

.... Second if you have any two superfunctions f and g, then
\( g^{-1}(f(x)) \) (if g is suitably injective) is always a 1-periodic function, which follows from the definition of Abel and superfunction.

This 1-periodic function interests me. \( \theta(z)=g^{-1}(f(z))-z \), where g(z) is an entire function, and f(z) is some other super function for the same base.

We already discussed base eta=e^(1/e), where g is the upper superfunction (cheta), and f is the regular \( \text{sexp}_\eta(z) \) function. Theta is a 1-cyclic function defined for imag(z)>=0 as long as z isn't an integer. This is similar to the case for bases>eta, where f=sexp(z), which is tetration, and is defined from g(z), the regular superfunction, by a theta(z). Here theta(z) is the Kneser Riemann mapping function, with singularities at the integers. Theta(z) quickly decays to zero as imag(z) increases. So, theta(z) is defined if imag(z)>=0 and z is not an integer. The Schwarz reflection property is used to define sexp(z) for imag(z)<0.

For bases<eta, a different more complicated theta(z) function is involved. Take g as the upper superexponential, and f as the sexp function, with f(0)=1, f(-1)=0, with b=sqrt(2). \( \theta(z)=g^{-1}(f(z))-z \) Here, once again theta(z) is not defined if z is an integer. But since f is imaginary periodic, with a period of \( 2\pi i/(\ln(\ln(2)))\approx17.143 \), then there is another set of singularities for theta(z) at imag(z)=Period=~17.143I. So theta(z) is not defined outside of the area between these two lines, the real axis, and imag(z)=~17.143I. And theta(z) has singularities at integer values of z and integer values of z+Period*I.

This is much different than the standard Kneser mapping theta(z) which decays to zero as imag(z) increases. But it is possible to imagine for bases like b=sqrt(2), another theta(z) function which does decays to zero as imag(z) increases, which results in another different b=sqrt(2) superfunction with sexp like properties. See also, http://math.eretrandre.org/tetrationforu...hp?tid=515

I'm sure that theta(z) is also relevant for other superfunctions, other than exponentiation.
- Sheldon