User:Fephisto/sandbox
)^{2}dxdy}}{\int _{0}^{m}\int _{0}^{n}\left[interp2(1:m,1:n,{\mbox{data table here}},x,y)\right](y,x)dxdy}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/082d0ddb01cb39efc8b5eed4086d5ab54149fb40)
Convention on Cluster Munitions
Convention on Certain Conventional Weapons
Environmental Modification Convention
Ottawa Treaty
Geneva Conventions
Hague Conventions of 1899 and 1907
User:Fephisto/Islands of Micronesia
User:Fephisto/Islands of Oceania
User:Fephisto/Islands of Southeast Asia
User:Fephisto/Territorial Disputes between NATO Members
User:Fephisto/Bridge Inspection
User:Fephisto/Householder QR Decomposition
User:Fephisto/Refrigerant pressure table
User:Fephisto/Backwards stability
User:Fephisto/Backwards stable
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.[1]
Inverse
[edit]In 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 . [2]
Convergence
[edit]By an old result of Euler, repeated exponentiation convergence for real values inbetween and .[3]
Calculation of Iterated Exponential
[edit]In certain situations, one may calculate the iterated exponential, and certain constants remain of mathematical interest.
Connection to Lambert's Function
[edit]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, . [4]
Iteration on the Complex Plane
[edit]The function may also be extended to the complex plane, where such a map tends to display interesting fractal properties.[5]
Of particular interest is evaluation of the constant
Which does indeed converge [6] 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>
- ↑ Finch, S. R. (2003). Mathematical constants. Cambridge university press. pp.448-<452/ref>
Background
Analytic Properties of Iterated Exponentials
Minimum and Injectivity
We note that is concave up with minimum or </math>=(0.3678794412...,0.6922006276...)</math> <ref>De Villiers, J. M., & Robinson, P. N. (1986). The interval of convergence and limiting functions of a hyperpower sequence. American Mathematical Monthly, 13-23. - ↑ 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 (syntax error): {\displaystyle 𝑖^{𝑖𝑖 \cdots}} . Proceedings of the American Mathematical Society, 17(1), 67.
