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 The Generalized Gaussian Method (GGM) tommy1729 Ultimate Fellow Posts: 1,605 Threads: 363 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 Image(s) « 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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