Definition:Tetration

Definition
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 $b$.