User:Fephisto/sandbox

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

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/Voluntary childlessness

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 $x^{x}$. Such systems have induced some interesting mathematical constants and interesting fractal properties based on its generalization to the complex plane.

Inverse

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

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

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

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

Convergence

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

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

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

for such $z$ where such a process converges,

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

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

Namely, that

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

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

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.

Of particular interest is evaluation of the constant

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

Which does indeed converge and has been evaluated as

$~=0.43828+0.36059i$

<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>