11/20/2007, 09:31 PM
Aha!
\( \ln\left(1-\frac{-z}{\hspace{5} e^{-z} \hspace{5}}\right) \)
\( \ln\left(1-\left(-ze^{z}\right)\right) \)
\( \ln\left(1+ze^{z}\right) \)
So this is related to my previous findings! When z is the additive inverse of a fixed point, we get:
\(
\begin{eqnarray}
1+ze^{z}|_{\small z=-c_k} & = & -c_k e^{-c_k} \\
& = & 1+\frac{-c_k}{c_k} \\
& = & 1-1 \\
& = & 0
\end{eqnarray}
\)
Therefore, at the additive inverses of the fixed points, we'll get logarithmic singularities.
I'm not sure about the bases though, and there are a few other oddities I need to figure out, but it's cool to see even this level of relationship.
\( \ln\left(1-\frac{-z}{\hspace{5} e^{-z} \hspace{5}}\right) \)
\( \ln\left(1-\left(-ze^{z}\right)\right) \)
\( \ln\left(1+ze^{z}\right) \)
So this is related to my previous findings! When z is the additive inverse of a fixed point, we get:
\(
\begin{eqnarray}
1+ze^{z}|_{\small z=-c_k} & = & -c_k e^{-c_k} \\
& = & 1+\frac{-c_k}{c_k} \\
& = & 1-1 \\
& = & 0
\end{eqnarray}
\)
Therefore, at the additive inverses of the fixed points, we'll get logarithmic singularities.
I'm not sure about the bases though, and there are a few other oddities I need to figure out, but it's cool to see even this level of relationship.
~ Jay Daniel Fox
