Iterated Exponential Constants

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Euler (Le Lionnais 1983) and Eisenstein (1844) showed that the function $h(x)=x^{x^{x^{\mathinner{\mkern1mu\raise1pt\vbox{\kern7pt\hbox{.}}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}}}}$, where $x^{x^x}$ is an abbreviation for $x^{(x^x)}$, converges only for $e^{-e}\leq x\leq e^{1/e}$, that is, 0.0659...$\leq x\leq$ 1.44466.... The value it converges to is the inverse of $x^{1/x}$, which has a closed form expression in terms of Lambert's W-Function,

$$h(z)={W(-\ln z)\over -\ln z}$$ (1)

(Corless et al.). Knoebel (1981) gives

$$h(z)=1+\ln x+{3^2(\ln z)^2\over 3!}+{4^3(\ln z)^3\over 4!}+\ldots$$ (2)

(Vardi 1991). A Continued Fraction due to Khovanskii (1963) is

$x^{1/x}\!=1+{2(x-1)\over\strut\displaystyle x^2+1-{\strut\displaystyle (x^2-1)(... ...\strut\displaystyle (9x^2-1)(x-1)^2\over\strut\displaystyle 7x(x+1)-\ldots}}}}.$
	(3)

The function $g(x)=x^{{(1/x)}^{{(1/x)}^{\mathinner{\mkern1mu\raise1pt\vbox{\kern7pt\hbox{.}}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}}}}$ converges only for $e^{-1/e}\leq x\leq e^e$, that is, $0.692\ldots\leq x \leq 15.154\ldots.$ The value it converges to is the inverse of $x^x$.

Some interesting related integrals are

$$\int_0^1 x^x\,dx=\sum_{n=1}^\infty {(-1)^{n+1}\over n^n}=0.7834305107\ldots$$ (4)

$$\int_0^1 x^{-x}\,dx = \sum_{n=1}^\infty {1\over n^n}=1.2912859971\ldots$$ (5)

(Spiegel 1968, Abramowitz and Stegun 1972).

See also Lambert's W-Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.

Baker, I. N. and Rippon, P. J. ``A Note on Complex Iteration.'' Amer. Math. Monthly 92, 501-504, 1985.

Barrows, D. F. ``Infinite Exponentials.'' Amer. Math. Monthly 43, 150-160, 1936.

Creutz, M. and Sternheimer, R. M. ``On the Convergence of Iterated Exponentiation, Part I.'' Fib. Quart. 18, 341-347, 1980.

Creutz, M. and Sternheimer, R. M. ``On the Convergence of Iterated Exponentiation, Part II.'' Fib. Quart. 19, 326-335, 1981.

de Villiers, J. M. and Robinson, P. N. ``The Interval of Convergence and Limiting Functions of a Hyperpower Sequence.'' Amer. Math. Monthly 93, 13-23, 1986.

Eisenstein, G. ``Entwicklung von $\alpha^{\alpha^{\alpha^{\mathinner{\mkern1mu\raise1pt\vbox{\kern7pt\hbox{.}}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}}}}$.'' J. Reine angew. Math. 28, 49-52, 1844.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/itrexp/itrexp.html

Khovanskii, A. N. The Application of Continued Fractions and Their Generalizations to Problems in Approximation Theory. Groningen, Netherlands: P. Noordhoff, 1963.

Knoebel, R. A. ``Exponentials Reiterated.'' Amer. Math. Monthly 88, 235-252, 1981.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 22 and 39, 1983.

Mauerer, H. ``Über die Funktion $x^{x^{\mathinner{\mkern1mu\raise1pt\vbox{\kern7pt\hbox{.}}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}}}$ für ganzzahliges Argument (Abundanzen).'' Mitt. Math. Gesell. Hamburg 4, 33-50, 1901.

Spiegel, M. R. Mathematical Handbook of Formulas and Tables. New York: McGraw-Hill, 1968.

Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 12, 1991.