# Difference between revisions of "Base e1e"

Map of $$~f\!=\!\eta\!=\!\exp_{\exp(1/\mathrm e)}~$$; here $$~u+\mathrm i v=f(x\!+\!\mathrm i y)~$$
Map of $$~f\!=\!\log_{\exp(1/\mathrm e)}~$$; here $$~u+\mathrm i v=f(x\!+\!\mathrm i y)~$$
Thick green curve: $$y=\eta^x$$; thin red curve: $$y=(\sqrt{2})^x$$

Base e1e refers to the value of base $$b= \eta =\exp(1/\mathrm e)\approx 1.4446678610$$

and corresponding exponential, SuperExponential (in particular, the tetration) and the inverse functions. In future, this may refer also to the highest Ackermann functions to this base.

The specific value of $$b$$ corresponds to the single real fixed point of the exponential. For this case, the special formalism of exotic iteration had been developed. [1]

The special name, for example, Henryk constant, is required for the base $$~b=\exp(1/\mathrm e)\approx 1.4446678610~$$ , and notation $$\eta$$ is suggested for this value of $$b$$.

## Exponent and logarithm to base e1e

The complex maps of $$f=\exp_\eta$$ and $$f=\log_\eta$$ are shown in figures at right with lines $$u+\mathrm i v = f(x+\mathrm i y)$$ in the $$x$$,$$y$$ plane. These pictures look similar to those for the case $$b=\sqrt{2}\approx 1.414$$, see article Base sqrt2. and those for the case $$b=3/2=1.5$$, see article Base 1.5.

## Fixed points

Values of the fixed points and behaviour of the superfunctions and the Abel functions are pretty different for the similar values of base $$b$$ mentioned above.

At base $$b=\sqrt{2}\approx 1.414$$, there exist two real fixed points, $$L_1=2$$ and $$L_2=4$$.

At base $$b=\eta=\exp(1/\mathrm e)\approx 1.444$$, there exist two real fixed points, $$L=\mathrm e \approx 2.71$$.

At base $$b=1.5$$, there exist no real fixed points, and the superexponential is supposed to approach the complex fixed points $$L$$ and $$L^*$$ at the infinity; correspondently, the AbelExponential, the ArcTetration $$\mathrm{ate}$$ has branch points at these values.

The explicit plot of exponential to base $$\eta$$

## Superexponentials to base $$\eta$$

For the base $$b=\eta=\exp(1/\mathrm e)$$, the asymptotic expansion of the superexponential can be written in the following form:

$$\displaystyle \tilde F(z)=\mathrm e\cdot\left(1-\frac{2}{z}\left( 1+\sum_{m=1}^{M} \frac{P_{m}\big(-\ln(\pm z) \big)}{(3z)^m} +\mathcal{O}\!\left(\frac{|\ln(z)|^{m+1}}{z^{m+1}}\right) \right) \right)$$

where $$P$$ are polynomials;

$$P_{1}(t)=t$$
$$P_{2}(t)=t^{2}+t+1/2$$
$$P_{3}(t)=t^{3}+\frac{ 5}{ 2}t^{2}+\frac{ 5}{2}t +\frac{ 7}{10}$$
$$P_{4}(t)=t^{4}+\frac{13}{ 3}t^{3}+\frac{ 45}{6}t^{2}+\frac{53}{10}t +\frac{ 67}{60}$$
$$P_{5}(t)=t^{5}+\frac{77}{12}t^{4}+\frac{101}{6}t^{3}+\frac{83}{ 4}t^{2}+\frac{653}{60}t+\frac{2701}{1680}$$
...

More polynomials $$P$$ can be calculated substituting representation $$\tilde F$$ into the transfer equation

$$\tilde F(z\!+\!1)=\exp_\eta(\tilde F(z))$$

and performing the asymptotic analysis at large $$|z|$$.

## References

1. http://www.ams.org/journals/mcom/0000-000-00/S0025-5718-2012-02590-7/S0025-5718-2012-02590-7.pdf
http://mizugadro.mydns.jp/PAPERS/2012e1eMcom2590.pdf H.Trappmann, D.Kouznetsov. Computation of the Two Regular Super-Exponentials to base exp(1/e). Mathematics of Computation. Math. Comp., v.81 (2012), p. 2207-2227. ISSN 1088-6842(e) ISSN 0025-5718(p)