real slog developed at a fixed point

[Kouznetsov]

Dear Henryk. I already have expessed some doubts, but now I see, my misunderstanding begins even earlier.
I do not understand even the title: how the slog can be real at the fixed point?

On the other hand, I suggest the asymptotic development of sexp at large values of the imaginary part of the argument.
Let \( \varepsilon=\varepsilon(z)=\exp(\el z +R) \), where \( \el \approx 0.3+1.3 i \) is fixed point of logarithm and \( R \) is some complex constant. Then, equation \( f(z+1)=exp(f(z)) \) allows the asumptotic expansion, using \( \varepsilon \) as small parameter:
\( \mathrm{sexp}(z)=\sum_{n=0}^N a_n \varepsilon^n +\mathcal{O}(\varepsilon^{N+1}) \).
The first coefficients of this expansion are:
\( a_0=\el \),
\( a_1=1 \),
\( a_2=\frac{1/2}{\el-1}\approx -0.151314 - 0.2967488 \mathrm{i}~,~ \)
\( a_3=\frac{a_2+1/6}{\el^2-1}\approx -0.36976 + 0.98730 \mathrm{i} \)
\( a_4= a4=(a2/2.+a_2^2/2.+a_3+1./24.)/(\ell^3-1.)
\approx 0.025811 -0.017387 i \).
I evaluate the parameter \( R\approx 1.077961437527-0.946540963949 i \), fitting tetration sexp at large values of the imaginary part, with the approximation at \( N=4 \). I plot the deviation of tetration from the asymptoric approximation with \( N=0,1,2,3 \) in the figiures below.
   
The four plots correspond to the four asumptotic approximations. The deviations
\( f=\mathrm{slog}(z)-L \),
\( f=\mathrm{slog}(z)-\Big(L+\varepsilon(z)\Big) \),
\( f=\mathrm{slog}(z)-\Big(L+\varepsilon+a_2 \varepsilon^2 \Big) \),
\( f=\mathrm{slog}(z)-\Big(L+\varepsilon+a_2 \varepsilon^2 + a_3 \varepsilon^3\Big) \)
are shown in the complex plane with lines of constant phase and constant modulus.
Levels \( \arg(f)=-2,-1 \) are shown with thick red lines.
Level \( \arg(f)=0 \) is shown with thick black lines.
Levels \( \arg(f)=1,2 \) are shown with thick blue lines.
Levels \( \arg(f)=\pm \pi \) are shown with scratched lines. (these lines reveal the step of sampling used by the plotter).

Levels \( |f|=\exp(-0.,\exp(-0.6),\exp(-0.4),\exp(-0.2) \) are shown with thin red lines.
Levels \( |f|=\exp(0.2),\exp(0.4),\exp(0.6),\exp(-0. \) are shown with thin blue lines.
Levels \( |f|=\exp(3), \exp(2), \exp(1),\exp(0), \exp(-\!1), \exp(-\!2), \exp(-\!3), \exp(-\!4), \exp(-\!5), \exp(-\!5), \exp(-\!7), \exp(-\! \) are shown with thin thick black lines.
Level \( |f|=\exp(-10) \) is shown with thick red line.
Levels \( |f|= \exp(-12),\exp(-14), \exp(-16),\exp(-1 \) are shown with thick black lines.
Level \( |f|=\exp(-20) \) is shown with thick red line.
Levels \( |f|= \exp(-22),\exp(-24), \exp(-26),\exp(-2 \) are shown with thick black lines.
Level \( |f|=\exp(-30) \) is shown with thick green line.
Level \( |f|=\exp(-31) \) is shown with thick black line.
The plotter tried to draw also
Level \( |f|=\exp(-32) \) with thick black line and
Level \( |f|=\exp(-33) \) with thin dark green line, which are seen a the upper left hand side corners of the two last pictures, but the precision of evaluation of tetration is not sufficient to plot the smooth lines; for the same reason, the curve for
\( |f|=\exp(-31) \) in the last picture, in the upper right band side looks a little bit irreguler; also, the pattenn in the upper left corner of the last two pictures looks chaotic; the plotter cannot distinguish the function from its asymptotic aproximation.

The figure indicates that, at \( \Im(z)>4 \), \( \Re(z)<4\Im(z) - 25 \), the asymptotic approximation
\( \mathrm{sexp}(z)\approx L+\varepsilon+a_2 \varepsilon^2 + a_3 \varepsilon^3 \)
gives at least 14 correct significant figures. At large values of the imaginary part, this approximation is more precise than the evaluation of tetration through the contour integral.

Questions.
1. Could anybody confirm this result with some independent way of evaluation of the superexponentiation?
2. Can we invert the series and get the expansion for the slog? Is it the same as Henryk has posted?
This may be not so straightforward. I see, there is different behavior in the left hand side of the figure and in the right hand side. This may mean that we should add more exponential terms, which are not just integer power of \( \varepsilon \). Similarly, we may need some additional logarithmic terms in the expansion of slog for the robust approximation.