The Generalized Gaussian Method (GGM) tommy1729 Ultimate Fellow Posts: 1,859 Threads: 402 Joined: Feb 2009 09/25/2021, 12:24 PM The Gaussian method can be easily generalized. suppose we use f(s) = exp( g(s) f(s-1) ) , then we are bounded in the sense that g(s) cannot grow to fast towards 1 as Re(s) goes to +oo. The reason is, if g(s) grows like O(exp(-exp(s)) ) then the (complex) argument (theta) gives us trouble. With erf(s) we are close to 1 + exp(-s^2) and because s^2 puts the imaginary line at 45° that is ok. With 1 + exp(-exp(s)) however the complex argument (theta) gives us issues. 1 + exp(-exp(s)) goes to 1 fast for positive real s , BUT because of the complex argument ( theta ) this does not hold for non-real s even if their real parts are large. So we look for functions g(s) between 1 + exp(-s^2) and 1 + exp(-exp(s)). This is cruxial to understand ! So how do we do that ? For starters it is also known that functions below O(exp(s)) can be completely defined by the value at 0 and its zero's. And we want the zero's to be close to the imag axis. This results in my generalized gaussian method. see pictures !! Regards Tom Marcel Raes tommy1729 Attached Files Thumbnail(s)         tommy1729 Ultimate Fellow Posts: 1,859 Threads: 402 Joined: Feb 2009 10/26/2021, 10:41 PM (This post was last modified: 10/28/2021, 12:11 PM by tommy1729.) 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 tommy1729 Ultimate Fellow Posts: 1,859 Threads: 402 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 »

 Possibly Related Threads… Thread Author Replies Views Last Post Pictures of some generalized analytical continuations Caleb 18 305 03/17/2023, 12:56 AM Last Post: tommy1729 greedy method for tetration ? tommy1729 0 75 02/11/2023, 12:13 AM Last Post: tommy1729 tommy's "linear" summability method tommy1729 15 504 02/10/2023, 03:55 AM Last Post: JmsNxn another infinite composition gaussian method clone tommy1729 2 149 01/24/2023, 12:53 AM Last Post: tommy1729 Semi-group iso , tommy's limit fix method and alternative limit for 2sinh method tommy1729 1 240 12/30/2022, 11:27 PM Last Post: tommy1729 [MSE] short review/implem. of Andy's method and a next step Gottfried 4 576 11/03/2022, 11:51 AM Last Post: Gottfried Some "Theorem" on the generalized superfunction Leo.W 59 22,489 09/18/2022, 11:05 PM Last Post: tommy1729 Is this the beta method? bo198214 3 625 08/18/2022, 04:18 AM Last Post: JmsNxn Describing the beta method using fractional linear transformations JmsNxn 5 816 08/07/2022, 12:15 PM Last Post: JmsNxn The Etas and Euler Numbers of the 2Sinh Method Catullus 2 555 07/18/2022, 10:01 AM Last Post: Catullus

Users browsing this thread: 1 Guest(s)