11/22/2007, 09:08 PM
Interestingly, the singularity is repeated at every tetrate of e (0, 1, e, e^e, etc.), and the other singularities are scaled copies of the one at the origin, plus the constant n for the n-th tetrate.
The scaling factor is simple to calculate. The singularity at the n-th tetrate is \( {}^{n} e \) times bigger than the scaling factor at the (n-1)-th tetrate. This can be seen by considering:
\(
\begin{eqnarray}
{\Large e}^{\left({}^{n-1}e+\delta\right)}
& = & {\Large e}^{\left({}^{n-1}e\right)} {\Large e}^{\delta} \\
& = & {}^{n}e \left(1+\delta+\mathcal{O}(\delta^2)\right) \\
& = & {}^{n}e + {}^{n}e \delta
\end{eqnarray}
\)
The scaling factor is simple to calculate. The singularity at the n-th tetrate is \( {}^{n} e \) times bigger than the scaling factor at the (n-1)-th tetrate. This can be seen by considering:
\(
\begin{eqnarray}
{\Large e}^{\left({}^{n-1}e+\delta\right)}
& = & {\Large e}^{\left({}^{n-1}e\right)} {\Large e}^{\delta} \\
& = & {}^{n}e \left(1+\delta+\mathcal{O}(\delta^2)\right) \\
& = & {}^{n}e + {}^{n}e \delta
\end{eqnarray}
\)
~ Jay Daniel Fox
