Arguments for the beta method not being Kneser's method

[JmsNxn]

And the approximation which I found helpful to find such points is that very nearby these bad points we will have
\( \beta(z-1)+z=2n\pi i \)



I have computed the value of the singularities for larger values.  For example here is n=1000, n=-1000, where Imag(z) is getting arbitrarily close to the real axis.  This is also expected from the plots above. 

Code:

n=1000   5.5396 + 0.0809*I;   5.5396028393500 + 0.080893337890853*I;
n=-1000  5.6147 + 0.2071*I;   5.6146550235446 + 0.20711039961980*I;

So sadly, the Beta method turns out to be another nowhere analytic tetration function.

I'm very confused by all of this. I think I'll have to wait for our zoom call.

All of the numbers you've posted evaluate small, but non-zero; no matter the depth of iteration I invoke. There also seems to be no branch-cuts after your singularities... how does that work?

If what you are saying is true; then the taylor series I construct, are asymptotic series..? That would be weird as hell. How would that even work?

For example, your values:

Code:

Abel_N(5.5396028393500 + 0.080893337890853*I,1,25,1E4)
%144 = -0.001640578620915541720546392322283529845247050181666649897021944942580396018970544869081577543125402054 + 0.0001330499723511153381095033009004166676723112972582777521859374100086903405093786723964333456249613833*I

Abel_N(5.6146550235446 + 0.20711039961980*I,1,25,1E4)
%145 = -0.001494252208177497453195498036687343307300150852039900705890034951897316546716029377330118581351032966 + 0.0003140841206461058389577698504561473693145040276189261025881173189719449811296695643286504880324094698*I

And stays there, before iterations over flow. Where I've used the code:

Code:

/*count is a limit on how many iterations; LIM is a limiter to quit before beta overflows*/

tau(z,y,{count=25},{LIM=1E4}) ={
    if(count>0 && real(Const(beta(z,y))) <= LIM,
        count--;
        log(1+tau(z+1,y,count,LIM)/beta(z+1,y)) - log(1+exp(-z*y)),
        -log(1+exp(-z*y))
    );
}

/*iferr is just primitively catching overflows and printing 1E100000*/

Abel_N(z,y,{count=25}, {LIM = 1E4}) = {
    if(real(Const(z)) <= 0,
        iferr(beta(z,y) + tau(z,y,count,LIM),E,1E100000,errname(E) == "e_OVERFLOW"),
        iferr(exp(Abel_N(z-1,y,count,LIM)),E,1E100000,errname(E)=="e_OVERFLOW")
    );
}

If this is a nowhere analytic tetration; it's probably the weirdest fucking nowhere analytic function! Lmao!!! Definitely one for the history books! If I can grab an asymptotic series at every point... and it's not analytic? That makes this even more interesting tbh! Definitely doesn't help dethrone Kneser or anything, but it's certainly really really fucking cool!

I'm not convinced yet; but I'll pivot my thesis if need be, lol!

Great work again, Sheldon--thank you. I'm still not convinced. I think we're at a standoff momentarily. But I'll give you the benefit of the doubt momentarily, and try to corroborate what you are investigating.

Looks like I got my weekend planned out, lol.

Regards, James