The Generalized Gaussian Method (GGM)
#1
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


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#2
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.











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 











regards

tommy1729
Tom Marcel Raes
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#3
(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.











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 











regards

tommy1729
Tom Marcel Raes

A further idea is to generalize like this 
 for positive odd 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.











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 











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
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