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Block Cholesky

Block Cholesky Decomposition

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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

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In fact, does have an inverse

which is well-defined for

This has induced interest in the function , which has similar limiting properties to . [1]

Convergence

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By an old result of Euler, repeated exponentiation convergence for real values inbetween and .[2]

Calculation of Iterated Exponential

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In certain situations, one may calculate the iterated exponential, and certain constants remain of mathematical interest.

Connection to Lambert's Function

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If 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

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The 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>

  1. ^ De Villiers, J. M., & Robinson, P. N. (1986). The interval of convergence and limiting functions of a hyperpower sequence. American Mathematical Monthly, 13-23.
  2. ^ L. Euler, De formulis exponentialibus replicatis, Leonhardi Euleri Opera Omnia, Ser. 1, Opera Mathematica 15 (1927) 268-297
  3. ^ 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.
  4. ^ Baker, I. N., & Rippon, P. J. (1985). A note on complex iteration. American Mathematical Monthly, 501-504.
  5. ^ 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.