Tommy's Gaussian method.

[JmsNxn]

I've attached here a quick patchwork code of Tommy's Gaussian method. I'm currently compiling some graphs. I wrote this a little fast and loose, so it's only accurate for about 15 digits or so.

Code:

/* This is a quick write up of Tommy's method. It's very slow and I haven't optimized it yet. It's only good for about 15 digits.
It's really rough around the edges but it's getting the job done. This seems a lot simpler to code than the beta method.
I do still believe this is the beta method though; I'm making graphs to double check though
*/

Err(z) = {
    (1+intnum(X=0,z,exp(-X^2)))/2;
}

/*these if statements are basically just to catch overflow errors and to exit if the values get too large*/

Tom(z) = {
    my(val=0);
    for(i=0,20,
        if(abs(val) <= 1E4,
            val = exp(Err(z-21+i)*val),
            if(abs(val)<=1E8,
                val = exp(Err(z-21+i)*val)),
                return(val);
            );
        );
    val;
}

/* This turns the multiplicative case of Tommy into the additive case I'm more used to*/

Conv_Tom(z) = {
    Err(z)*Tom(z);
}

/*This is the error term between the converted Tommy function and tetration*/

tau(z,{count=0}) = {
    if(real(Tom(z)) < 1E4 && count < 6,
        count++;
        log(1+tau(z+1,count)/Conv_Tom(z+1)) + log(Err(z+1)),
        log(Err(z+1));
    );
}

/*This is the tetration function, it's not normalized yet, though */

Tet_Tom(z) = {
       Conv_Tom(z) + tau(z);
}

This is an example of it converging on the real line. I'll update this post with contour plots. Unfortunately the code is really slow, so it might be a day.

Unnormalized, the domain is \( X \in [-1,3] \)
   


A slightly more accurate representation is given here, where you can discern it's a tetration function. Here, \( X = [-1.5,2] \)--again, it's not normalized.

   

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Here's a good amount of evidence that Tommy's method is holomorphic. Again, this code isn't perfect; alors, there are some exponent over flows in the process, which force the couple of hairs you see. This is over \( 0 \le \Re(z) \le 1 \) and \( -0.5 \le \Im(z)\le 0.5 \)--again, unnormalized. Please, ignore the hairs; we need a matrix add on to avoid this.

   

Tommy's method is absolutely holomorphic!