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Cmplex map of combination of natural ArcTetration and growing superexpnential to base \(\sqrt{2}\):

\(f(z)=\mathrm{ate}\Big(\mathrm{SuExq2}(z)\Big)\)

The  map  is shown in the top picture with

lines \(u=\Re \big(f(x\!+\!\mathrm i y)\big)\)  and

lines \(v=\Im \big(f(x\!+\!\mathrm i y)\big)\) 

in the \(x,y\) plane.

\vskip 12pt
\sx{1.4}{\bf References}
\vskip 6pt

\url{https://mizugadro.mydns.jp/BOOK/468.tex}
D.Kouznetwov. Superfunctions. Lambert Academic Publishing, 2020 
\vskip 5pt

\url{http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html} \\
\url{https://mizugadro.mydns.jp/PAPERS/2010sqrt2.pdf}
D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.
\vskip 5pt

\url{http://www.vmj.ru/articles/2010_2_4.pdf}
D.Kouznetsov. Tetration as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45, In Rusian.  English version:
\url{https://mizugadro.mydns.jp/PAPERS/2009vladie.pdf} 
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