Arguments for the beta method not being Kneser's method

[sheldonison]

James, vacationing; can't add any more calculations for a few days.  The key question seems to be whether the resultant Tetration function converges everywhere  in the complex plane if 0<|Im(z)|<pi, especially near where Beta(z+2)=1, |Beta(z+1)| is small.

I HAVE A STUPENDOUS UPDATE FOR YOU SHELDON

I've gotten 1E-18 in the taylor series in a neighborhood of 1!!!!!!

That is I've gotten exp(func(-0.5)) - func(0.5) = 1E-18 where func is a 100 term taylor series!!!!!!!!

I AM SO EXCITED!!!!

I think I fixed my taylor series. 64 bit pari is so much goddamn better!

Rregards, James

Have a great vacation! Have a pina colada on me!

Hi James,
Sounds like you're making progress in understanding the Beta function!  One of the problems with iterating logarithms is knowing how many 2pi i multiple's are required.  This can often be resolved by comparing the logarithm with the function with one less iteration, and picking the 2n*Pi*I branch which is closest.  

The key question seems to be whether the resultant Tetration function converges ... especially near where Beta(z+2)=1, |Beta(z+1)| is small.

But unfortunately I found a singularitiy.  Consider z=5.3136167434369 + 0.80386188968627*I; where beta(z+1,1)=1, and beta(z) is small.  So this location is a logarithmic singularity for your Tetration function.   I found this singularity by looking nearby the zeros of \( \beta(z-1,1)+z=0 \), since looking directly for zeros of \( \beta(z)-\ln(1+\exp(z)) \) gets many nearly zero results that are false positives, where beta(z-1) has a negative real part.