“Iterated Exponential Constants”的概念、定义、习题解题方法及翻译-数学辞典

单词

Iterated Exponential Constants

释义

Iterated Exponential Constants

N.B. A detailed on-line essay by S. Finchwas 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 ofLambert'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).

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 .'' 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.

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