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 Conjectures jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 10/10/2007, 07:43 AM andydude Wrote:Super-logarithm Derivative From my approximations, $\text{slog}_e'(0) \approx 0.916$, and ${}^{i}{e} \approx 0.786 + i 0.916$. My first conjecture is that $\text{Im}({}^{i}{e}) = \text{slog}_e'(0)$, and my second conjecture is that $\text{slog}_e'(0) = 0.915965594177\cdots$ otherwise known as Catalan's constant.In computing the solutions (for base e) to moderately large systems with your proposed slog, I think the first derivative is closer to 0.9159460564995..., which puts Catalan's constant out of the running (or puts your slog out of the running?). Quote:Super-logarithm Equilibrium My experiments with the super-logarithm have shown that there is a line in which the real part of the complex-valued superlog do not depend on the imaginary part of the input, in other words, my conjecture is that there exist functions f(b) and g(a, b) such that: $\text{slog}_b(f(b) + i a) = \text{slog}_b(f(b)) + i g(a, b)$ Could you provide a bit more info on this one? Again using calculations based on your solution with base e, I've seen a line near z=0.5 (somewhere between 0.45 and 0.5), running almost a unit's length "up" and "down" in the imaginary direction, for which the slog has nearly fixed real part. But it's not quite exact, and as z approaches either primary fixed singularity, the slog eventually gets "sucked in". But maybe you had something else in mind? ~ Jay Daniel Fox « Next Oldest | Next Newest »

 Messages In This Thread Conjectures - by andydude - 10/09/2007, 06:57 PM RE: Conjectures - by bo198214 - 10/09/2007, 09:23 PM RE: Conjectures - by jaydfox - 10/10/2007, 07:43 AM RE: Conjectures - by jaydfox - 12/04/2007, 12:23 AM RE: Conjectures - by jaydfox - 12/04/2007, 05:21 AM RE: Conjectures - by andydude - 12/04/2007, 10:35 PM RE: Conjectures - by andydude - 01/09/2008, 10:58 PM RE: Conjectures - by andydude - 10/13/2007, 04:51 PM RE: Conjectures - by andydude - 10/13/2007, 05:13 PM

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