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Tetration is the superfunction of the exponential map. For a given base \(b\), the tetration \(\operatorname{tet}_b\) is defined as the function satisfying the transfer equation \[ \operatorname{tet}_b(z+1) = b^{\operatorname{tet}_b(z)} \] together with additional normalization and regularity conditions that select a unique solution among infinitely many possible superfunctions.

The name “tetration’’ reflects its role as the next operation after exponentiation within the hyperoperation hierarchy.

The most studied case is the real holomorphic tetration to base \(b=\mathrm e\), written simply as \[ \operatorname{tet}(z)=\operatorname{tet}_{\mathrm e}(z). \]

Real tetration to base \(b=\mathrm e\).
From: Fig.17.2, p.239 in Superfunctions.

Definition

Let \(T_b(z)=b^z\). A function \(F\) is a superexponential (a superfunction of \(T_b\)) if \[ F(z+1) = T_b(F(z)). \]

A tetration to base \(b\) is the unique real-holomorphic superexponential for which:

1. the limits

 \[
  \lim_{y\to\pm\infty} F(x+\mathrm i y)
  \]
 remain finite (regularity at the imaginary infinities),

1. the normalization

 \[
  F(0)=1
  \]
 holds.

These conditions remove the otherwise infinite freedom in solutions to the transfer equation.

Properties

For the principal tetration \(\operatorname{tet}_b\):

• **Transfer equation**

\[ \operatorname{tet}_b(z+1)=b^{\operatorname{tet}_b(z)}. \]

• **Derivative at the fixed point**

Let \(L\) satisfy \(L = b^L\). Then \[ \operatorname{tet}_b'(0)=L. \]

• **Analyticity**

For \(1 < b < \mathrm e^{1/\mathrm e}\), the tetration to base \(b\) extends to an entire function. For \(b > \mathrm e^{1/\mathrm e}\), branch singularities appear, and a global entire tetration is impossible.

Regularity in the base

The tetration depends smoothly on the base \(b\) for \[ 1 < b < \mathrm e^{1/\mathrm e}. \] Within this range, the exponential map \(z\mapsto b^z\) has two attracting fixed points, allowing construction of an entire Abel function and an entire tetration.

At the critical base, the two fixed points merge, and the dynamical structure changes.

Singularity at the critical base \(b = 1/\mathrm e\)

The point \[ b = \frac{1}{\mathrm e} \] is a branch point of the exponential map in the parameter \(b\). Although tetration is smooth for \(b>1/\mathrm e\), this regularity does *not* guarantee holomorphic extendability across the branch point.

Important observations:

• When the base parameter \(b\) is continued into the complex plane, the behavior around \(b=1/\mathrm e\) becomes path-dependent.
• Continuation of tetration above and below the branch point generally does not give the same result — the continuation is multivalued.
• The situation is analogous to the relation between the Exponential integral \(\operatorname{Ei}\) and \(E_1\): one can smoothly approach the singularity from either side, but no single global holomorphic branch exists.
• For tetration, the complication is deeper because the function depends on two variables \((b,z)\); nonetheless, \(b=1/\mathrm e\) remains a limiting point for the tetrations defined on both sides.

This is one of the important unresolved structural problems of tetration theory.

Complex structure

Ackermann-family functions.

Natural tetration (dashed) and other Ackermann-like functions.

From: Fig.19.7, p.266 in Superfunctions.

The complex analytic structure of tetration is governed by the dynamics of \(z\mapsto b^z\). For \(1<b<\mathrm e^{1/\mathrm e}\), the map has two attracting fixed points. This allows one to construct an Abel function that is analytic on the entire plane and thus an entire tetration.

At the critical base \(b=\mathrm e^{1/\mathrm e}\), the two fixed points collide (parabolic bifurcation). For \(b>\mathrm e^{1/\mathrm e}\), both fixed points become repelling, and the Abel function — hence tetration — develops branch points and cannot be entire.

Relation to superfunctions

A superfunction \(F\) of \(T_b\) satisfies \[ F(z+1)=b^{F(z)}. \]

Among the infinitely many such functions, the tetration is selected by:

• realness on the real axis,
• holomorphy in a vertical strip,
• regularity at \(+\mathrm i\infty\) and \(-\mathrm i\infty\),
• normalization at \(z=0\).

All other superexponentials differ by periodic or quasiperiodic distortions.

Inverse: Abel function

The inverse of tetration is the Abel function (ArcTetration), denoted \(\operatorname{ate}_b\). It satisfies \[ \operatorname{ate}_b(b^z) = \operatorname{ate}_b(z) + 1, \] with \[ \operatorname{ate}_b(1)=0. \]

For the natural exponential: \[ \operatorname{ate}(\exp(z)) = \operatorname{ate}(z) + 1. \]

Iterates

Tetration and ArcTetration provide analytic extensions of integer iteration. For the \(n\)-th iterate of the exponential: \[ \exp_b^n(z) = \operatorname{tet}_b\!\big(n + \operatorname{ate}_b(z)\big), \] valid for real or complex \(n\).

Special cases

Tetration near the critical base \(b=\mathrm e^{1/\mathrm e}\).
From: Fig.17.4, p.245 in Superfunctions.

Base \(\sqrt{2}\)

A classical example of an entire tetration, used extensively in early numerical work.

Base \(b= \mathrm e^{1/\mathrm e}\)

This “critical base’’ marks the transition where the fixed points of \(z\mapsto b^z\) merge. Properties include:

• extremely slow growth near the fixed point,
• non-entire analytic continuation,
• delicate dependence on initial conditions.

Base 2

An important superexponential used in comparisons with hyperoperators.

Natural tetration

The principal tetration \(\operatorname{tet}(z)\) is real-analytic on the real axis. As \(z\to -\infty\), it approaches the lower fixed point of \(\exp(z)\). As \(z\to +\infty\), it grows faster than any finite exponential tower.

Figures

Tetration and its inverse (ArcTetration).
From: Fig.14.4, p.203 in Superfunctions.

Complex structure of tetration.
From: Fig.18.3 (top), p.250 in Superfunctions.

Conceptual illustration

Heuristic “half-sheep’’ illustration.
From: Fig.15.6, p.218 in Суперфункции (2014).

This cartoon illustrates a philosophical point in tetration theory: heuristic assumptions that appear “obvious’’ (e.g., that the right side of a sheep has the same color as the left) may be false without rigorous proof.

Likewise, assumptions about tetration — such as smoothness in both arguments \((b,z)\), or holomorphic extendability across \(b=1/\mathrm e\) — require careful justification. The cartoon emphasizes that the behavior of superexponentials around their branch points is subtler than it may first appear.

References

D. Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020. All cited figures except the last are taken from this monograph.

D. Kouznetsov. Суперфункции. Lambert Academic Publishing, 2014. Figure 15.6 (the “half-sheep’’) is taken from this edition.

Keywords

«Base e1e», «Base sqrt2», «Superfunction», «Exponential», «Logarithm», «Tetration», «ate», «Abel function», «Transfer equation», «Iterates».