The Generalized Gaussian Method (GGM)

[tommy1729]

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