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Iterated exponentials are an example of an iterated function system based on . Such systems have induced some interesting mathematical constants and interesting fractal properties based on its generalization to the complex plane.Cite error: A <ref> tag is missing the closing </ref> (see the help page).
Inverse
editIn fact, does have an inverse
which is well-defined for
![{\displaystyle y\in [({\frac {1}{e}})^{\frac {1}{e}},e^{e}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f7212e387348e752fdaf275d427c46ac29f2c04)
This has induced interest in the function , which has similar limiting properties to . [1]
Convergence
editBy an old result of Euler, repeated exponentiation convergence for real values inbetween and .[2]
Calculation of Iterated Exponential
editIn certain situations, one may calculate the iterated exponential, and certain constants remain of mathematical interest.
Connection to Lambert's Function
editIf one defines
for such where such a process converges,
Then actually has a closed form expression in terms of a function known as Lambert's function which is defined implicitly via the following equation:
Namely, that
This can be seen by inputting this definition of into the other equation that satisfies, . [3]
Iteration on the Complex Plane
editThe function may also be extended to the complex plane, where such a map tends to display interesting fractal properties.[4]
Of particular interest is evaluation of the constant
Which does indeed converge [5] and has been evaluated as
<ref>Galidakis, I. N. (2004). On an application of Lambert's W function to infinite exponentials. Complex Variables, Theory and Application: An International Journal, 49(11), 759-780.</math>
- ^ De Villiers, J. M., & Robinson, P. N. (1986). The interval of convergence and limiting functions of a hyperpower sequence. American Mathematical Monthly, 13-23.
- ^ L. Euler, De formulis exponentialibus replicatis, Leonhardi Euleri Opera Omnia, Ser. 1, Opera Mathematica 15 (1927) 268-297
- ^ Corless, R. M., Gonnet, G. H., Hare, D. E., Jeffrey, D. J., & Knuth, D. E. (1996). On the Lambert W function. Advances in Computational mathematics, 5(1), 329-359.
- ^ Baker, I. N., & Rippon, P. J. (1985). A note on complex iteration. American Mathematical Monthly, 501-504.
- ^ Macintyre, A. J. (1966). Convergence of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 𝑖^{𝑖𝑖 \cdots}} . Proceedings of the American Mathematical Society, 17(1), 67.