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 Pictures of the Chi-Star JmsNxn Long Time Fellow Posts: 507 Threads: 89 Joined: Dec 2010 05/28/2017, 08:46 PM (This post was last modified: 05/28/2017, 09:38 PM by JmsNxn.) Very pretty pictures. I thought I might add an interesting quip about how to calculate this complex super function. I recently ran into this formula working on difference equations, and it turned out to be equivalent to solving for super functions. (Funny how the brain works, when working on two entirely different problems, they happen to be intensely related.) Define the sequence $\zeta_n = \Psi^{-1}(e^{-n} \Psi(x_0))$ where $x_0$ is fixed and is arbitrary, so long as $\log^{\circ n}(x_0) \to L$. We can write an entire super function of exponentiation as $F(z)=\frac{1}{\Gamma(Lz)} (\sum_{n=0}^\infty \zeta_n \frac{(-1)^n}{n!(n+Lz)} + \int_1^\infty (\sum_{n=0}^\infty \zeta_n \frac{(-t)^n}{n!})t^{Lz-1}\,dt)$ This expression converges FOR ALL complex values, which is REAL nice. Naturally $e^{F(z)} = F(z+1)$, but $F(z) \neq 0$, and is not real valued on the real line. I have equally so been wondering if this can somehow be morphed into a solution to tetration, but no luck so far. I'm still a little lost about how Kneser does it, but even how it looks, it doesn't seem like it'l apply in this scenario. EDIT: I thought I'd add that its periodic too. Since this super function looks locally like $F(z) = \Psi^{-1}(e^{Lz}\Psi(x_0))$, it follows that $F(z+\frac{2\pi i}{L}) = F(z)$ « Next Oldest | Next Newest »

 Messages In This Thread Pictures of the Chi-Star - by sheldonison - 05/28/2017, 01:13 AM RE: Pictures of the Chi-Star - by JmsNxn - 05/28/2017, 08:46 PM RE: Pictures of the Chi-Star - by sheldonison - 05/29/2017, 09:14 AM RE: Pictures of the Chi-Star - by sheldonison - 05/31/2017, 02:40 AM RE: Pictures of the Chi-Star - by JmsNxn - 05/29/2017, 05:07 PM RE: Pictures of the Chi-Star - by JmsNxn - 05/31/2017, 05:38 PM RE: Pictures of the Chi-Star - by sheldonison - 06/01/2017, 12:49 AM RE: Pictures of the Chi-Star - by JmsNxn - 06/01/2017, 07:29 AM RE: Pictures of the Chi-Star - by sheldonison - 06/01/2017, 01:41 PM RE: Pictures of the Chi-Star - by JmsNxn - 06/07/2017, 08:18 PM RE: Pictures of the Chi-Star - by sheldonison - 06/08/2017, 01:19 PM RE: Pictures of the Chi-Star - by JmsNxn - 06/08/2017, 09:20 PM RE: Pictures of the Chi-Star - by sheldonison - 06/09/2017, 01:11 PM RE: Pictures of the Chi-Star - by JmsNxn - 06/09/2017, 05:25 PM

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