05/23/2011, 09:01 PM
(This post was last modified: 05/23/2011, 10:37 PM by sheldonison.)
The recent posts on the Taylor series for the superfunctions of \( \eta=exp(1/e) \) reminded me that I want to post my theory, that as imag(z) increases, the lower and upper superfunctions at eta converge towards each other, plus a constant. Here, sexp(z) is the lower superfunction, with sexp(0)=1, and cheta(z) is the upper superfunction, normalized so that cheta(0)=2e.
\( \text{sexp}_{\eta}(z)=\text{cheta}(z+\theta(z)+ k )
\). Where \( \theta(z) \) is a 1-cyclic function, which quickly decays to zero as imag(z) increases. Then, the constant "k" is 5.0552093131039000 + 1.0471975511965977*I. And then we have for any real number x,
\( \lim_{z \to i\infty}\text{sexp}_{\eta}(x+z)=\text{cheta}(x+z+k) \)
- Sheldon
\( \text{sexp}_{\eta}(z)=\text{cheta}(z+\theta(z)+ k )
\). Where \( \theta(z) \) is a 1-cyclic function, which quickly decays to zero as imag(z) increases. Then, the constant "k" is 5.0552093131039000 + 1.0471975511965977*I. And then we have for any real number x,
\( \lim_{z \to i\infty}\text{sexp}_{\eta}(x+z)=\text{cheta}(x+z+k) \)
- Sheldon