Maps of tetration

$y\!=\!\mathrm {tet} _{b}(x)$ versus $x$ for various $b$ by

Base $b={\sqrt {2}}\approx 1.41$

Henryk base, $b=\exp(1/\mathrm {e} )\approx 1.44$

Base $b=1.5$

Binary tetration, $b=2$

Natural base, $b=\mathrm {e} \approx 2.71$

Sheldon base, $b=1.52598338517+0.0178411853321\,\mathrm {i}$

Article Maps of tetration collects some complex maps of tetration $\mathrm {tet} _{b}$ to different values of base $b$.

For several real values of base $b\!>\!1$ the real-real plots $y\!=\!\mathrm {tet} _{b}(x)$ is shown at the upper figure at right.

The complex maps correspond to the following values of base:

$b={\sqrt {2}}\approx 1.41$,
$b=\exp(1/\mathrm {e} )\approx 1.44$, 
$b=1.5$,
$b=2$,
$b=\mathrm {e} \approx 2.71$,
$b=1.52598338517+0.0178411853321\,\mathrm {i}$

Tetration is shown with lines of constant real part $u$ and lines of constant imaginary part $v$; $u\!+\!\mathrm {i} v=\mathrm {tet} _{b}(x\!+\!\mathrm {i} y)$

$b={\sqrt {2}}$

For this case, the regular iteration at fixed point $L=2$ is used. The evaluation is described in the Mathematics of Computation .

$b=\exp(1/\mathrm {e} )\approx 1.44$

For $b=\exp(1/\mathrm {e} )\approx 1.44$, the exotic iteration at fixed point $L=\mathrm {e} \approx 2.71$ is used. The evaluation is described in the Mathematics of Computation .

$b>\exp(1/\mathrm {e} )$

For $b>\exp(1/\mathrm {e} )\approx 1.44$, the Cauchi integral is used for evaluation. It is described in Mathematics of Computation .

Historically, evaluation for the case $b=\mathrm {e}$ was first to be reported. Namely for this case, the special algorithm fsexp.cin is loaded; it is described in Vladikavkaz Mathematical Jorunal .

Sheldon base $b=1.52598338517+0.0178411853321\,\mathrm {i} .$

Tetration to Sheldon base $b\!=\!1.52598338517\!+\!0.0178411853321\mathrm {i}$ is considered by the special request from Sheldon Levenstein. For this base, it was believed to be especially difficult to evaluate.

The evaluation uses almost the same algorithm of the Cauchi integral .

The small modification had been applied to the original algorithm; the condition $F(z^{*})=F(z)^{*}$ is suppressed at the numerical solving of the corresponding integral equation for values of superfunction along $\Im (z)=\mathrm {const}$. No difficulties, specific namely for this complex value of base $b$, had been detected.

Book

The maps are plotted using the conto.cin code in C++. The Latex code is used to add the labels. All the maps at right are supplied with generators; the colleagues may download the codes and reproduce (or even modify) them. If some generator does not work as expected, let me know and let us correct it.

The algorithms, used to evaluate the tetration to various bases, are described also in the Book Суперфункции, in Russian . For year 2014, the English version is not yet ready. The images above are prepared for the article "Holomorphic ackermanns" .

Content of the first version of this article is adopted from TORI, http://mizugadro.mydns.jp/t/index.php/Maps_of_tetration

Keywords

Ackermann Complex map Tetration