Definition:Tetration

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 $b$.