## 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 , where is an abbreviation for , converges only for , that is, 0.0659... 1.44466.... The value it converges to is the inverse of , which has a closed form expression in terms of Lambert's W-Function,

 (1)

(Corless et al.). Knoebel (1981) gives
 (2)

(Vardi 1991). A Continued Fraction due to Khovanskii (1963) is
 (3)

The function converges only for , that is, The value it converges to is the inverse of .

Some interesting related integrals are

 (4)

 (5)

(Spiegel 1968, Abramowitz and Stegun 1972).

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 .'' 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 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.