04/30/2009, 11:29 PM
(This post was last modified: 04/30/2009, 11:30 PM by BenStandeven.)
Let's see here.
The fixed point for base (eta + eps) is e + delta(eps), where delta satisfies:
\( [1/e + \ln(1 + \eps/\eta)] (e + \delta(\eps)) = 1 + \ln(1 + \delta(\eps)/e) \)
\( [1/e + \eps/\eta + O(eps^2)] (e + \delta(\eps)) = 1 + \ln(1 + \delta(\eps)/e) \)
\( [\eps/\eta + O(\eps^2)] (e + \delta(\eps)) = -(\delta(\eps)/e)^2/2 + O(\delta(\eps)^3) \)
So \( \delta(\eps) \approx \sqrt{\eps} \) and
\( 0 = e\eps/\eta + \delta(\eps) \eps/\eta + (\delta(\eps)/e)^2/2 + O((\eps)^{3/2} \)
\( \delta(\eps) = \frac{ - \eps/\eta + \sqrt{ \eps^2/\eta^2 - \frac{2 \eps}{e \eta} } } { 1/e^2 } + O(\eps^{3/2}) \)
\( \delta(\eps) = \frac{ - \eps/\eta + \sqrt{\frac{-2 \eps}{e \eta}} [1 - \frac{e \eps}{4 \eta} + O(\eps^2)] } { 1/e^2 } + O(\eps^{3/2}) \)
\( \delta(\eps) = -e^2 \eps/\eta + \sqrt{2/\eta} e^{3/2} \sqrt{-\eps} + O(\eps^{3/2}) \)
The fixed point for base (eta + eps) is e + delta(eps), where delta satisfies:
\( [1/e + \ln(1 + \eps/\eta)] (e + \delta(\eps)) = 1 + \ln(1 + \delta(\eps)/e) \)
\( [1/e + \eps/\eta + O(eps^2)] (e + \delta(\eps)) = 1 + \ln(1 + \delta(\eps)/e) \)
\( [\eps/\eta + O(\eps^2)] (e + \delta(\eps)) = -(\delta(\eps)/e)^2/2 + O(\delta(\eps)^3) \)
So \( \delta(\eps) \approx \sqrt{\eps} \) and
\( 0 = e\eps/\eta + \delta(\eps) \eps/\eta + (\delta(\eps)/e)^2/2 + O((\eps)^{3/2} \)
\( \delta(\eps) = \frac{ - \eps/\eta + \sqrt{ \eps^2/\eta^2 - \frac{2 \eps}{e \eta} } } { 1/e^2 } + O(\eps^{3/2}) \)
\( \delta(\eps) = \frac{ - \eps/\eta + \sqrt{\frac{-2 \eps}{e \eta}} [1 - \frac{e \eps}{4 \eta} + O(\eps^2)] } { 1/e^2 } + O(\eps^{3/2}) \)
\( \delta(\eps) = -e^2 \eps/\eta + \sqrt{2/\eta} e^{3/2} \sqrt{-\eps} + O(\eps^{3/2}) \)
