Definition:Tetration

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Definition

$y = \map {\operatorname {tet}_b} x$ versus $x$ for various $b$

$f = \map {\operatorname {tet}_b} x$ 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^{\paren { {}^{n - 1} x} } & : n > 0 \\ \end {cases}$

Using Knuth uparrow notation:

$x \uparrow \uparrow n := \begin {cases}

1 & : n = 0 \\ x \uparrow \paren {x \uparrow \uparrow \paren {n - 1} } & : n > 0 \\ \end {cases}$

Definition for base $b \ge \map \exp {1 / e}$

Let $b \in \R$ such that $b \ge \map \exp {\dfrac 1 e}$.

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

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

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

$\map {\operatorname {tet}_b} 0 = 1$

$\forall z \in C: \map {\operatorname {tet}_b} {z^*} = \map {\operatorname {tet}_b} z^*$

$\ds \forall x \in \R: \lim_{y \mathop \to +\infty} \map {\operatorname {tet}_b} {x + \mathrm i y} = L$

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

Definition for $0 < b < \map \exp {1 / e}$

Let $b \in \R$ such that $1 < b < \map \exp {\dfrac 1 e}$.

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

Let $T = \dfrac{2 \pi i} {\map \ln {L_1 \map \ln b} }$

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

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

$\map {\operatorname {tet}_b} 0 = 1$

$\forall z \in C: \map {\operatorname {tet}_b} {z^*} = \map {\operatorname {tet}_b} z^*$

$\forall z \in C: \map {\operatorname {tet}_b} z = \map {\operatorname {tet}_b} {z + T}$

$\ds \forall y \in \R: \lim_{x \mathop \to -\infty} \map {\operatorname {tet}_b} {x + \mathrm i y} = L_2$

$\ds \forall \epsilon \in \R_{>0}: \exists X \in \R$ such that:

$\forall x \in \R: x > X: \size {\map {\operatorname {tet}_b} {x + i y} - L_1} < \epsilon$

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