This is the user sandbox of Fephisto. A user sandbox is a subpage of the user's user page. It serves as a testing spot and page development space for the user and is not an encyclopedia article. Create or edit your own sandbox here. Other sandboxes: Main sandbox | Template sandbox Finished writing a draft article? Are you ready to request review of it by an experienced editor for possible inclusion in Wikipedia? Submit your draft for review!

${\displaystyle {\frac {\sqrt {\int _{0}^{m}\int _{0}^{n}(\left[interp2(1:m,1:n,{\mbox{data table here}},x,y)\right](y,x))^{2}dxdy}}{\int _{0}^{m}\int _{0}^{n}\left[interp2(1:m,1:n,{\mbox{data table here}},x,y)\right](y,x)dxdy}}}$

Convention on Cluster Munitions Convention on Certain Conventional Weapons Environmental Modification Convention Ottawa Treaty Geneva Conventions Hague Conventions of 1899 and 1907

User:Fephisto/Territorial Disputes between NATO Members

User:Fephisto/Bridge Inspection

User:Fephisto/Significant New Alternatives Policy

User:Fephisto/Article 5 of the North Atlantic Treaty

User:Fephisto/Voluntary childlessness

User:Fephisto/2022 Nord Stream pipeline sabotage

Block Cholesky

Block Cholesky Decomposition

Template:Aviation Briefing navbox

Template:NATO Militarised Interstate Conflicts

Iterated exponentials are an example of an iterated function system based on ${\displaystyle x^{x}}$. 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

In fact, ${\displaystyle f(x)=x^{x}}$ does have an inverse

${\displaystyle x=f^{-1}(y)=y^{({\frac {1}{y}})^{({\frac {1}{y}})^{\dots }}}}$

which is well-defined for ${\displaystyle y\in [({\frac {1}{e}})^{\frac {1}{e}},e^{e}]}$

This has induced interest in the function ${\displaystyle x^{\frac {1}{x}}}$, which has similar limiting properties to ${\displaystyle x^{x}}$. ^[1]

Convergence

By an old result of Euler, repeated exponentiation convergence for real values inbetween ${\displaystyle e^{-e}}$ and ${\displaystyle e^{\frac {1}{e}}}$.^[2]

Calculation of Iterated Exponential

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

Connection to Lambert's Function

If one defines

${\displaystyle h(z)=z^{z^{z^{\dots }}}}$

for such ${\displaystyle z}$ where such a process converges,

Then ${\displaystyle h(z)}$ actually has a closed form expression in terms of a function known as Lambert's function which is defined implicitly via the following equation:

${\displaystyle W(z)e^{W(z)}=z}$

Namely, that

${\displaystyle h(z)={\frac {W(-log(z))}{-log(z)}}}$

This can be seen by inputting this definition of ${\displaystyle h(z)}$ into the other equation that ${\displaystyle h(z)}$ satisfies, ${\displaystyle h(z)=z^{h(z)}}$. ^[3]

Iteration on the Complex Plane

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

${\displaystyle i^{i^{i^{\dots }}}}$

Which does indeed converge ^[5] and has been evaluated as

${\displaystyle ~=0.43828+0.36059i}$

<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>