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

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