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 The Generalized Gaussian Method (GGM) tommy1729 Ultimate Fellow Posts: 1,493 Threads: 356 Joined: Feb 2009 10/28/2021, 12:07 PM (This post was last modified: 10/28/2021, 12:12 PM by tommy1729.) (10/26/2021, 10:41 PM)tommy1729 Wrote: The related integral above is quite complicated. So I came up with the following simplification. A different method but very similar. n are integers larger than 0. m is going to +infinity. $f(s)=e^{t(s)*f(s-1)}$ $t(s)=(J(s)+1)/2$ $J(s) =Erf(s*p_7(s))$ $p_7(s)=\prod(1+s^2/n^7)$ $sexp(s+s_e)=\ln^{[m]}f(s+m)$ This has similar properties as the other generalized gaussian method and it should be easier to implement. call it GGM2 or so. For bases other than e ; take the base e^b then we get  $f_b(s)=e^{b*t(s)*f(s-1)}$ $t(s)=(J(s)+1)/2$ $J(s) =Erf(s*p_7(s))$ $p_7(s)=\prod(1+s^2/n^7)$ $sexp_b(s+s_b)=\ln_b^{[m]}f_b(s+m)$ regards tommy1729 Tom Marcel Raes A further idea is to generalize like this   for positive odd w ;  $t_w(s)=1+(J(s)-1)^w/2^w$ for instance w = 3 or w = 7. with w = 7 we get the case : n are integers larger than 0. m is going to +infinity. $f(s)=e^{t_w(s)*f(s-1)}$ $t_w(s)=1+(J(s)-1)^w/2^w$ $J(s) =Erf(s*p_7(s))$ $p_7(s)=\prod(1+s^2/n^7)$ $sexp(s+s_e)=\ln^{[m]}f(s+m)$ This has similar properties as the other generalized gaussian method and it should be easier to implement. call it GGM2 or so. For bases other than e ; take the base e^b then we get  $f_b(s)=e^{b*t_w(s)*f(s-1)}$ $t_w(s)=1+(J(s)-1)^w/2^w$ $J(s) =Erf(s*p_7(s))$ $p_7(s)=\prod(1+s^2/n^7)$ $sexp_b(s+s_b)=\ln_b^{[m]}f_b(s+m)$ Notice this latest new modifation does not change the range where we get close to 1 much , but is still getting faster to 1. regards tommy1729 Tom Marcel Raes ps : join " tetration friends " at facebook :p « Next Oldest | Next Newest »

 Messages In This Thread The Generalized Gaussian Method (GGM) - by tommy1729 - 09/25/2021, 12:24 PM RE: The Generalized Gaussian Method (GGM) - by tommy1729 - 10/26/2021, 10:41 PM RE: The Generalized Gaussian Method (GGM) - by tommy1729 - 10/28/2021, 12:07 PM

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