E489  De formulis exponentialibus replicatis
(On unravelling exponential formulas)
Summary:
The problem Euler addresses in this paper was posed by Condorcet.
Euler takes r positive, and α real, then asks for what values the sequence
r, r^{α}, r^{rα}, r^{rrα}, ...
converges. The critical value seems to be at r = e^{1/e} = 1.44467.
According to the
records, it was presented to the St. Petersburg Academy on June 12, 1777.
Publication:

Originally published in Acta Academiae Scientarum Imperialis Petropolitinae 1, 1778, pp. 3860

Opera Omnia: Series 1, Volume 15, pp. 268  297
Documents Available:
 Original Publication: E489
 The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E489 include:
 Baker IN, Rippon PJ., “A note on complex iteration.” American Mathematical Monthly, 92 (7), pp. 501504 (1985).
 Baker IN, Rippon PJ., “Iteration of exponential functions.” Annales Academiae Scientiarum FennicaeMathematica, 9 (1), pp. 4977 (1984).
 Knoebel RA., “Exponentials reiterated.” American Mathematical Monthly, 88 (4), pp. 235252 (1981).
 Rippon PJ., “Infinite exponentials.” Mathematical Gazette, 67 (441), pp. 189196 (1983).
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