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 how many superfunctions? [was superfunctions of eta converge towards each other] sheldonison Long Time Fellow Posts: 641 Threads: 22 Joined: Oct 2008 05/31/2011, 07:38 PM (This post was last modified: 05/31/2011, 07:57 PM by sheldonison.) (05/28/2011, 09:18 AM)bo198214 Wrote: .... 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

 Messages In This Thread how many superfunctions? [was superfunctions of eta converge towards each other] - by tommy1729 - 05/26/2011, 10:09 PM RE: how many superfunctions? [was superfunctions of eta converge towards each other] - by bo198214 - 05/27/2011, 09:33 AM RE: how many superfunctions? [was superfunctions of eta converge towards each other] - by tommy1729 - 05/27/2011, 10:48 PM RE: how many superfunctions? [was superfunctions of eta converge towards each other] - by bo198214 - 05/28/2011, 09:18 AM RE: how many superfunctions? [was superfunctions of eta converge towards each other] - by sheldonison - 05/31/2011, 07:38 PM RE: how many superfunctions? [was superfunctions of eta converge towards each other] - by tommy1729 - 05/28/2011, 12:12 PM RE: how many superfunctions? [was superfunctions of eta converge towards each other] - by bo198214 - 05/28/2011, 12:25 PM RE: how many superfunctions? [was superfunctions of eta converge towards each other] - by tommy1729 - 05/29/2011, 05:27 PM RE: how many superfunctions? [was superfunctions of eta converge towards each other] - by bo198214 - 05/29/2011, 09:20 PM

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