Definition:Tetration

Definition for base $b \ge \exp (1/e)$
Let $b \in \R$, $b \ge \exp (1/e)$.

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

Let $C = \C \backslash \{x \in \R: x \le -2 \}$.

Let $\operatorname{tet}_b: C \mapsto \C$ be the superfunction of $z \mapsto b^z$ such that:
 * $\operatorname{tet}_b (0) = 1$
 * $\forall z \in C: \operatorname{tet}_b(z^*) = \operatorname{tet}_b(z)^*$
 * $\displaystyle \forall x \in \R: \lim_{y \to +\infty} \operatorname{tet}_b (x + \mathrm i y) = L $

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

Definition for $0<b<\exp(1/e)$
Let $b \in \R$, $1< b < \exp (1/e)$.

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

Let $\displaystyle T=\frac{2\pi \mathrm i} {\ln(L_1 \, \ln(b))} $

Let $C = \C \backslash \{x+ T m, x \in \R: x \le -2, m \in \mathbb Z \}$,

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 $


 * $\displaystyle \forall \varepsilon \in \R: \varepsilon\!>\!0~ \exists~ X\!\in\! \R$ such that $|\operatorname{tet}_b (x\! +\! \mathrm i y) - L_1| < \varepsilon ~\forall x\!\in\!\R: x\!>\!X$

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