Arguments for the beta method not being Kneser's method

[sheldonison]

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I can see it getting very large, but beta is non zero. This can only blow up if \( -\tau(z) = \beta(z) \); they're both small, so I see it in the realm of possibility....
So these would be our bad points; where these functions intersect. Interesting; this will definitely be helpful.

Hey James,
Correct, these are the problem points where \( -\tau(z) = \beta(z) \) which is also where \( \beta(z+1)=1 \) and \( \beta(z) \) is small.
And the approximation which I found helpful to find such points described in post#36 is that very nearby these bad points we will have\( \beta(z-1)+z=2n\pi i \)
This is only an approximation.  Here are the 11 singularities associated with n= -5 ... +5, along with the approximation nearby I used to find the bad point. As the absolute value of n gets arbitrarily large, the singularities will approach arbitrarily near the real axis.

Code:

 n  2nPi approximation   singularity nearby where beta(z)+tau(z)=0            
-5  5.4610 + 0.5136*I;   5.4609932449971 + 0.51357174630384*I;
-4  5.4563 + 0.5354*I;   5.4563694801667 + 0.53541269413988*I;
-3  5.4502 + 0.5646*I;   5.4502899247509 + 0.56458029140716*I;
-2  5.4407 + 0.6070*I;   5.4407276325107 + 0.60701009407168*I;
-1  5.4182 + 0.6780*I;   5.4183215379239 + 0.67799916571271*I;
 0  5.3132 + 0.8037*I;   5.3136167434369 + 0.80386188968627*I;
 1  5.0437 + 0.7376*I;   5.0435559986965 + 0.73816709432084*I;
 2  5.0027 + 0.5490*I;   5.0023867230678 + 0.54911331696556*I;
 3  5.0309 + 0.4485*I;   5.0307340294544 + 0.44846615796161*I;
 4  5.0620 + 0.3901*I;   5.0618719624426 + 0.39007387338227*I;
 5  5.0889 + 0.3518*I;   5.0888072383831 + 0.35176025961591*I;

These two graphs showing where the singularities are go from real(3) ... real(6) and imag(-0.5) to imag(1.5)
The first graph is \( \beta(z-1)+z\pm 2n\pi i \) where n is chosen to minimize the imaginary portion of \( \beta(z-1)+z \).  Where this function is zero, nearby there will be a singularity where \( \beta(z)=-\tau(z) \).  
   
The 2nd graph is of \( 100\cdot(\beta(z)+\tau(z)) \)  I multiplied by 100 because otherwise its hard to see the zeros since the green portion of this graph is very small in magnitude.  Each of these zero corresponds to a singularity.  I realize more time could be spent explaininig the approximation from post #36, but its hard to do that online... What I can say is the approximation was extremely helpful in computing these zeros/singularities where \( \beta(z)=-\tau(z) \).    I view this equivalently as \( \beta(z)-\ln(1+\exp(-z))=0 \)
   
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.

edit: explaining the approximation.  This is by no means a proof, but it is an explanation.  I haven't tried to rigorously prove the approximation since I was mostly interested in finding the singularities.
let's suppose \( \beta(z-1)+z=2n\pi i \)
Then \( \beta(z-1)=-z+2n\pi i \)
Then \( \beta(z)=\frac{\exp(\beta(z-1))}{1+\exp(-z))} \)
If real(z) is large enough then this approximation is pretty good, and the denominator is approximately 1.
Then \( \beta(z)\approx\exp(\beta(z-1))\approx\exp(-z+2n\pi i)\approx\exp(-z) \)
now \( \tau(z)=-\ln(1+\exp(-z)) \) and if real(z) is large enough then \( \tau(z)\approx-\exp(-z) \)
So if we start with this approximation, then we might expect that nearby we will find an exact value where the following is exactly true:  \( \tau(z)=-\beta(z) \)