File:TetrationAsymptoticExpansion00.jpg

Tertation tet is solution of equations ${\displaystyle \mathrm {tet} (z\!+1\!)=\exp {\big (}\mathrm {tet} (z){\big )}}$ ${\displaystyle \mathrm {tet} (0)=1}$, holomorphyc in the complex plane except the cut at the part of the real axis. At large values of the imaginary part of the argument, tetratiomn can expanded asumptoticaly, ${\displaystyle \mathrm {tet} (z)=\sum _{n=0}^{N}a_{n}\varepsilon ^{n}+{\mathcal {(}}O)\!{\big (}\varepsilon ^{N\!+\!1}{\big )}}$ where ${\displaystyle \varepsilon =\varepsilon (z)=\exp(Lz+R)}$ ; ${\displaystyle L\approx 0.2+1.3\mathrm {i} }$ is fixed point of logarithm, and ${\displaystyle R\approx 1.077961437526-0.946540963949\mathrm {i} }$ is universal constant. First coefficients are: ${\displaystyle a_{0}=L}$, ${\displaystyle a_{1}=1}$, ${\displaystyle a_{2}={\frac {1/2}{L-1}}\approx -0.151314-0.2967488\mathrm {i} ~,~}$ ${\displaystyle a_{3}={\frac {a_{2}+1/6}{L^{2}-1}}\approx -0.36976+0.98730\mathrm {i} }$
, The four pictures show the deviation of tetration from the approcimation at ${\displaystyle N\!=\!0}$, ${\displaystyle N\!=\!1}$, ${\displaystyle N\!=\!2}$, and ${\displaystyle N\!=\!3}$; id est, ${\displaystyle f=\mathrm {tet} (z)-\sum _{n=0}^{N}a_{n}\varepsilon ^{n}}$. The distribution of ${\displaystyle f}$ in the complex ${\displaystyle z}$-plane is shown with lines of constant phase and lines of constant modulus of ${\displaystyle f}$.

Levels ${\displaystyle \arg(f)=-2,-1}$ are shown with thick red lines.

Level ${\displaystyle \arg(f)=0}$ is shown with thick black lines.

Levels ${\displaystyle \arg(f)=1,2}$ are shown with thick blue lines.

Levels ${\displaystyle \arg(f)=\pm \pi }$ are shown with scratched lines.

Levels ${\displaystyle |f|=\exp(-0.8),\exp(-0.6),\exp(-0.4),\exp(-0.2)}$ are shown with thin red lines.

Levels ${\displaystyle |f|=\exp(0.2),\exp(0.4),\exp(0.6),\exp(-0.8)}$ are shown with thin blue lines.

Levels ${\displaystyle |f|=\exp(3),\exp(2),\exp(1),\exp(0),\exp(-\!1),\exp(-\!2),\exp(-\!3),\exp(-\!4),\exp(-\!5),\exp(-\!5),\exp(-\!7),\exp(-\!8)}$ are shown with thin thick black lines.

Level ${\displaystyle |f|=\exp(-10)}$ is shown with thick red line.

Levels ${\displaystyle |f|=\exp(-12),\exp(-14),\exp(-16),\exp(-18)}$ are shown with thick black lines.

Level ${\displaystyle |f|=\exp(-20)}$ is shown with thick red line.

Levels ${\displaystyle |f|=\exp(-22),\exp(-24),\exp(-26),\exp(-28)}$ are shown with thick black lines.

Level ${\displaystyle |f|=\exp(-30)}$ is shown with thick green line.

Level ${\displaystyle |f|=\exp(-31)}$ is shown with thick black line.

The plotter tried to draw also

Level ${\displaystyle |f|=\exp(-32)}$ with thick black line and

Level ${\displaystyle |f|=\exp(-33)}$ with thin dark green line, but they happened a littel bit scratched.

In the upper left corner of the last two pictures; the plotter could not distinguish the tetration from its asymptotic approximation.

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