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 Superlog with exact coefficients Gottfried Ultimate Fellow Posts: 765 Threads: 119 Joined: Aug 2007 06/11/2008, 07:18 AM (This post was last modified: 06/11/2008, 12:44 PM by Gottfried.) andydude Wrote:This means that $\beta(x) = \text{slog}_{(b^{1/b})}(x)$ which relates back to the super-logarithm as follows: $\text{slog}_{(e^{a(e^{-a})})}(x) = C + \frac{1}{\ln(a)}\left( (x(e^{-a})-1) + \frac{a(x(e^{-a})-1)^2}{4(1-a)} + \frac{a^2(1+5a)(x(e^{-a})-1)^3}{36(a-1)^2(a+1)} - \frac{a^4(2+a+3a^2)(x(e^{-a})-1)^4}{32(a-1)^3(a+1)(1+a+a^2)} + \cdots \right)$ Hmm, at least it looks somehow similar to the "regular"-formula. If I replace a by u, and introduce t for exp(u) so that $\text{slog}_{(e^{a(e^{-a})})}(x) = \text{slog}_{(e^{u/t})}(x)$ and $(x(e^{-a})-1) = \frac{x}{t}-1 = x'$ and factorize the denominators differently, for instance $\hspace{24} (a-1)^3(a+1)(1+a+a^2) = (a-1)(a-1)(a-1)*(a+1)*(1+a+a^2) = (a-1)(a^2-1)(a^3-1)$ to get a more familiar looking formula for me, this is then, using "rsdxplog" as rslog for the dxp-function: $\text{slog}_{e^{u/t}}(x) = \text{rsdxplog}_t(x') = C + \frac{1}{\ln(u)}\left( (x' - \frac{ux'^2}{4(u-1)} \hspace{12} + \frac{u^2(1+5u)x'^3}{36(u-1)(u^2-1)} \hspace{12} - \frac{u^4(2+u+3u^2)x'^4}{32(u-1)(u^2-1)(u^3-1)} \hspace{12} + \cdots \right)$ where especially the denominators-products are the same as in my Ut-formulae, and also the numerators look very familiar. I'll see, whether we have the same coefficients later today. I had the rsdxplog as logarithm of the Schroeder-function, assuming x' as h'th (continuous) iteration of $\text{dxp}_t^{{^o}h}(1)$ then $ \\ \vspace{12}\hspace{24} x' = \text{dxp}_t^{{^o}h}(1) = \sigma_t^{-1}(u^h \sigma_t(1)) \\ \vspace{12}\hspace{24} \sigma_t(x') = u^h \sigma_t(1) \\ \vspace{12}\hspace{24} \frac {\sigma_t(x')}{\sigma_t(1)} = u^h \\ \vspace{12}\hspace{24} \text{rsdxplog}(x')= h = \log_u(\sigma_t(x')) - \log_u(\sigma_t(1)) \\ \vspace{12}\hspace{24} \text{rsdxplog}(x')= h = C + \log_u(\sigma_t(x')) =C + \frac1{ln(u)}\log(\sigma_t(x'))$ where I got the coefficients of the sigma-function by the eigenmatrices of Ut - and the structure of these coefficients look very similar to yours above Gottfried Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread Superlog with exact coefficients - by andydude - 06/11/2008, 05:46 AM RE: Superlog with exact coefficients - by andydude - 06/11/2008, 06:02 AM RE: Superlog with exact coefficients - by Gottfried - 06/11/2008, 07:18 AM RE: Superlog with exact coefficients - by Gottfried - 06/13/2008, 06:38 AM RE: Superlog with exact coefficients - by bo198214 - 06/20/2008, 01:26 PM RE: Superlog with exact coefficients - by Gottfried - 03/10/2009, 09:59 PM

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