real slog developed at a fixed point

[Kouznetsov]

\( r_0 \) has to be chosen such that \( \text{slog}(0)=-1 \), i.e
\( -1=\text{slog}(0)=\frac{1}{\el}\left(\log(-\el) + r_0+\sum_{n=1}^\infty r_n (-\el)^n\right) \)
i.e.
\( r_0=-\el-\log(-\el)-\sum_{n=1}^\infty r_n (-\el)^n \)
*headscratch, is that correct?*

Well, if \( r_0 \approx -1.0779614375277+0.9465409639482 i \)
then, can we plot the pics?

P.S. Henryk, I have computational indication, that the series above, even if converges, does not converge to the values of the slog function.
I calculated the jump of the slog function along its cut. The code:
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
#include <complex.h>
#define z_type complex<double>
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "f4natu.cin"
#include "f4d1natu.cin"
#include "g4natu.cin"
main(){ int m,n; DB x,y; z_type z,c,d, cu,cd;
z_type L=z_type(.31813150520476413, 1.3372357014306895);

DO(m,31)
{ x=Re(L)-.1*m;
z=z_type(x,Im(L)+1.e-14); cu=G4natu(z);
z=z_type(x,Im(L)-1.e-14); cd=G4natu(z); d=cu-cd; c=d*L;
printf("%5.2f %6.3f %15.12f %15.12f %15.12f %15.12f\n",.1*m,x,Re(d),Im(d), Re©,Im©);
}
}
Below, is the output.
Zeroth column: distance from the fixed point L.
First column: x, Real part of the argument of slog
next two columns: Real and imaginary parts of the jump evaluated as
d= slog(L - x + i *1.e-14) - slog(L - x - i *1.e-14).
next two columns: Real and imaginary parts of d/L .
0.00 0.318 2.204629320832 0.510377772480 0.018866665677 3.110476285219
0.10 0.218 4.446950517660 1.057939822526 0.000000161107 6.283184982867
0.20 0.118 4.446949163231 1.057934723290 0.000006549101 6.283181549449
0.30 0.018 4.446950722702 1.057908310360 0.000042365531 6.283175232044
0.40 -0.082 4.446968541263 1.057833866255 0.000147583492 6.283175376644
0.50 -0.182 4.447023057115 1.057676880467 0.000374853702 6.283198335062
0.60 -0.282 4.447137525037 1.057397004506 0.000785529681 6.283262368294
0.70 -0.382 4.447333728986 1.056950220488 0.001445403878 6.283382603147
0.80 -0.482 4.447628633453 1.056291468498 0.002420128961 6.283567390167
0.90 -0.582 4.448032384297 1.055377506697 0.003770757173 6.283816540166
1.00 -0.682 4.448547681229 1.054169596920 0.005549949441 6.284121339465
1.10 -0.782 4.449170275877 1.052635649573 0.007799265569 6.284465898277
1.20 -0.882 4.449890217828 1.050751616712 0.010547697790 6.284829260148
1.30 -0.982 4.450693461716 1.048502095034 0.013811375676 6.285187742835
1.40 -1.082 4.451563518823 1.045880232677 0.017594216201 6.285517117241
1.50 -1.182 4.452482939627 1.042887105440 0.021889229526 6.285794391492
1.60 -1.282 4.453434513703 1.039530746214 0.026680198603 6.285999106707
1.70 -1.382 4.454402151619 1.035824994068 0.031943498780 6.286114150167
1.80 -1.482 4.455371465040 1.031788294715 0.037649886408 6.286126149438
1.90 -1.582 4.456330089184 1.027442545470 0.043766145990 6.286025536119
2.00 -1.682 4.457267802332 1.022812043024 0.050256535271 6.285806370905
2.10 -1.782 4.458176496661 1.017922564733 0.057084004499 6.285466012314
2.20 -1.882 4.459050048272 1.012800594128 0.064211190742 6.285004696498
2.30 -1.982 4.459884125528 1.007472688141 0.071601203094 6.284425079632
2.40 -2.082 4.460675965723 1.001964975531 0.079218222344 6.283731779706
2.50 -2.182 4.461424141950 0.996302771896 0.087027941621 6.282930942304
2.60 -2.282 4.462128335191 0.990510295126 0.094997874414 6.282029845292
2.70 -2.382 4.462789121224 0.984610465224 0.103097554447 6.281036550200
2.80 -2.482 4.463407777924 0.978624773560 0.111298649224 6.279959602927
2.90 -2.582 4.463986115531 0.972573208132 0.119575005978 6.278807783003
3.00 -2.682 4.464526330455 0.966474223171 0.127902645797 6.277589898421

In the first row, the last number pretends to be pi\approx 3.14, but my algorithm, evaluating slog, gives only 14 digits. As the distance from the branchpoint becomes macroscopic (for ex., 0.1, id est, much larger than the error of evaluation of slog), the jump of slog is similar to \( 2\pi i /L \) , but it grow with increase of distance from the branchpoint. The up to last colmn can be interpreted as indication of error of the approximation with log+polynomial, because log has constsant jump.
For this reason I expect that the series has radius of approximation zero.
(however, the series still may be very useful for the precise numerical evaluation in vicinity of the branchpoint)

Can anybody suggest a holomorphic function with some branchpoint and similar behavior of the jump along the cutline?