Definition:Tetration

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Definition

$y = \operatorname{tet}_b \left({x}\right)$ versus $x$ for various $b$
$f = \operatorname{tet}_b \left({x}\right)$ in the $x, b$ plane with levels $f = \text{const}$.

Definition for Integers

For all $x \in \R$, $n \in \Z_{\ge 0}$:

${}^n x := \begin{cases} 1 & : n = 0 \\ x^{ \left({{}^{n - 1} x}\right)} & : n > 0 \\ \end{cases}$


Using Knuth uparrow notation:

$x \uparrow \uparrow n := \begin{cases} 1 & : n = 0 \\ x \uparrow \left({x \uparrow \uparrow \left({n - 1}\right)}\right) & : n > 0 \\ \end{cases}$


Definition for base $b \ge \exp \left({1 / e}\right)$

Let $b \in \R$ such that $b \ge \exp \left({\dfrac 1 e}\right)$.

Let $L \in \C$ be a fixed point of $\log_b$ such that $\Im \left({L}\right) \ge 0$.

Let $C = \C \setminus \left\{ {x \in \R: x \le -2}\right\}$.


Let $\operatorname{tet}_b: C \mapsto \C$ be the superfunction of $z \mapsto b^z$ such that:

$\operatorname{tet}_b \left({0}\right) = 1$
$\forall z \in C: \operatorname{tet}_b \left({z^*}\right) = \operatorname{tet}_b \left({z}\right)^*$
$\displaystyle \forall x \in \R: \lim_{y \to +\infty} \operatorname{tet}_b \left({x + \mathrm i y}\right) = L$


Then the function $\operatorname{tet}_b$ is called tetration to base $b$.


Definition for $0 < b < \exp \left({1 / e}\right)$

Let $b \in \R$ such that $1 < b < \exp \left({\dfrac 1 e}\right)$.

Let $L_1, L_2 \in \R: L_1 < L_2$ be the fixed points of $\log_b$.

Let $T = \dfrac{2 \pi i} {\ln \left({L_1 \ln \left({b}\right)}\right)}$

Let $C = \C \setminus \left\{ {x + T m, x \in \R: x \le -2, m \in \Z}\right \}$


Let $\operatorname{tet}_b: C \mapsto \C$ be the superfunction of $z \mapsto b^z$ such that:

$\operatorname{tet}_b (0) = 1$
$\forall z \in C: \operatorname{tet}_b(z^*) = \operatorname{tet}_b(z)^*$
$\forall z \in C: \operatorname{tet}_b(z) = \operatorname{tet}_b(z+T)$
$\displaystyle \forall y \in \R: \lim_{x \to -\infty} \operatorname{tet}_b (x + \mathrm i y) = L_2 ~~ $
$\displaystyle \forall \varepsilon \in \R_{>0}: \exists X \in \R$ such that:
$\forall x \in \R: x > X: \left|{\operatorname {tet}_b \left({x + i y}\right) - L_1}\right| < \varepsilon$


Then the function $\operatorname{tet}_b$ is called tetration to base $b$.


Sources