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 superfunctions of eta converge towards each other sheldonison Long Time Fellow Posts: 626 Threads: 22 Joined: Oct 2008 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 « Next Oldest | Next Newest »

 Messages In This Thread superfunctions of eta converge towards each other - by sheldonison - 05/23/2011, 09:01 PM RE: superfunctions of eta converge towards each other - by bo198214 - 05/23/2011, 09:49 PM RE: superfunctions of eta converge towards each other - by sheldonison - 05/23/2011, 11:26 PM RE: superfunctions of eta converge towards each other - by bo198214 - 05/24/2011, 01:06 PM RE: superfunctions of eta converge towards each other - by sheldonison - 05/24/2011, 02:18 PM RE: superfunctions of eta converge towards each other - by bo198214 - 05/24/2011, 02:27 PM RE: superfunctions of eta converge towards each other - by sheldonison - 05/30/2011, 06:06 AM RE: superfunctions of eta converge towards each other - by sheldonison - 05/25/2011, 04:03 AM RE: superfunctions of eta converge towards each other - by sheldonison - 05/27/2011, 09:36 AM RE: superfunctions of eta converge towards each other - by tommy1729 - 05/25/2011, 11:04 PM RE: superfunctions of eta converge towards each other - by bo198214 - 05/25/2011, 11:15 PM RE: superfunctions of eta converge towards each other - by tommy1729 - 05/26/2011, 12:10 PM RE: superfunctions of eta converge towards each other - by tommy1729 - 06/06/2011, 10:42 PM

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