Tetration

Abstract: The Lambert W function has utility for solving various exponential and logarithmic equations arranged in the form of $g(x)e^{g(x)}$. Using the Lambert W function and tetration, a variety of categorized inversion formulas are presented. Related techniques are then used to derive polar forms of exponential and related functions.

Abstract: For a positive integer k let φk be the k-fold composition of the Euler function φ. In this paper, we study the size of the set {φk(n) ≤ x} as x tends to infinity.

Abstract: The author makes use of infinite compositions and a limiting function to construct a $\mathcal{C}^\infty$ tetration function $\mathcal{F}(t) = e \tet t$. As a tetration function, $\mathcal{F}$ satisfies $e^{\mathcal{F}(t)} = \mathcal{F}(t+1)$. Of it, $\mathcal{F}$ takes $(-2,\infty) \to \mathbb{R}$ bijectively with strictly monotone growth, and is continuously differentiable here. We then iterate this construction to derive arbitrary hyper-operations $e\up^k t$. These hyper-operations are $\mathcal{C}^\infty$ strictly monotone bijections of $(\alpha_k,\infty) \to \mathbb{R}$ for $k$ even ($1-k > \alpha_k \ge -k$), and $\mathcal{C}^\infty$ strictly monotone bijections of $\mathbb{R} \to (\alpha_k, \infty)$ for $k$ odd. These hyper-operations satisfy the functional equation $e \up^{k-1} (e \up^k t) = e \up^k (t+1)$ with the initial conditions $e \up^1 t = e^t$ and $e \up^k 0 = 1$.