Arguments for the beta method not being Kneser's method

[sheldonison]

For James,

I started meeting with James on a zoom call, more or less weekly, trying to understand his latest Tetration solution.  Here is a challenge problem for James.  Calculate the value for Abel_N(1+I,1).  I figure a single point as a challenge problem should help clarify the problems in general, and a single point allows one to focus.

After yesterday's zoom call, I started working with Jame's Abel_T.gp program, and I wanted to share my observations.

• The beta(z,1) function appears very well behaved in the complex plane, and matches to precision a Taylor series generated by sampling a unit circle, and also matches the functions iterative definition.   The results are accurate to 112 decimal digits with 240 sample points.
  
• The Abel_N(z,1) function does not match its Taylor series generated by sampling around a unit circle with 32 sample points, except at those 32 sample points.  
  

Here, I sampled 32 points around a unit circle for Abel_N(z,1) centered at z=1.  Granted, 32 sample points isn't particularly large, but its quicker that way, and easier to see the problems.  This Taylor series, will match Abel(N(z) at the 32 sample points around a unit circle exactly, but the difference is ... clearly not an analytic function, and adding more sample points is not going to help.  I also posted the 32 term Taylor series.  which matches the Abel_N(z,1) function at the 32 equally spaced sample points.  The graph is from 1+exp(0*I) ... 1+exp(Pi*I), or half of a unit circle graphing Abel_N(z,1)-TaylorSeries.

So, perhaps James can make some headway in figuring out the Abel_N(1+I,1) function which is approximated by 0.334 + 0.832i; the Taylor series gives 0.344+0.811i.

update: There is a discontinuity between Abel_N(1+exp(2.3826*I)) and Abel_N(1+exp(2.3827*I)).  Those two points could also be studied to see what causes the discontinuity, and to study the iterated function's convergence.

This post should have been added to this thread but I don't know how to move this post: 
https://math.eretrandre.org/tetrationfor...23#pid9723
   

Code:

{Abel_N=
        0.26962349367025
+x^ 1*  0.98289395290899
+x^ 2* -0.12839966506604
+x^ 3*  0.21830648113188
+x^ 4* -0.089448736367909
+x^ 5*  0.074647906187944
+x^ 6* -0.036017423946514
+x^ 7*  0.018234953617508
+x^ 8* -0.0058405324292364
+x^ 9*  0.0048183497091155
+x^10* -0.010002486474801
+x^11*  0.016389477074026
+x^12* -0.019181228205727
+x^13*  0.016131660699365
+x^14* -0.0075339894316277
+x^15* -0.0030394488907054
+x^16*  0.010082804929620
+x^17* -0.011107672494816
+x^18*  0.0074702878952322
+x^19*  0.00050519581149936
+x^20* -0.0091471972435540
+x^21*  0.011952484076219
+x^22* -0.0096472370827766
+x^23*  0.0055591676270766
+x^24*  0.0016849792938102
+x^25* -0.0099505377559867
+x^26*  0.011385235421743
+x^27* -0.0070217176746849
+x^28*  0.0027394592867243
+x^29*  0.0030327763362493
+x^30* -0.010678678609549
+x^31*  0.012604188471810
}