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added first part of validation

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Martin Thoma 2013-12-18 13:37:03 +01:00
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@ -205,12 +205,40 @@ $t$:
&= 0 &= 0
\end{align} \end{align}
\textbf{Case 2.2:} TODO \textbf{Case 2.2:}
\todo[inline]{calculate...}
\[x = \frac{(1+i \sqrt{3})a}{\sqrt[3]{12} \cdot t} \[x = \frac{(1+i \sqrt{3})a}{\sqrt[3]{12} \cdot t}
-\frac{(1-i\sqrt{3}) t}{2\sqrt[3]{18}}\] -\frac{(1-i\sqrt{3}) t}{2\sqrt[3]{18}}\]
\textbf{Case 2.3:} TODO \begin{align}
x^3 &= \underbrace{\left (\frac{(1+i\sqrt{3})a}{\sqrt[3]{12} \cdot t} \right)^3}_{=: \raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {1}}}}
\underbrace{- 3 \left(\frac{(1+i\sqrt{3})a}{\sqrt[3]{12} \cdot t} \right)^2 \left(\frac{(1-i\sqrt{3})t}{2 \sqrt[3]{18}} \right)}_{=: \raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {2}}}}\\
&\hphantom{{}=}+ \underbrace{3 \left(\frac{(1+i\sqrt{3})a}{\sqrt[3]{12} \cdot t} \right) \left(\frac{(1-i\sqrt{3})t}{2 \sqrt[3]{18}}\right)^2}_{=: \raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {3}}}}
\underbrace{- \left(\frac{(1-i\sqrt{3})t}{2 \sqrt[3]{18}}\right)^3}_{=: \raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {4}}}}
\end{align}
Now simplify the summands:
\begin{align}
\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {1}}} &=
\frac{a^3(1+3i\sqrt{3} - 3 \cdot 3 - \sqrt{27} i)}{12 t^3}\\
&= \frac{a^3((3\sqrt{3}- \sqrt{27})i - 8)}{12 t^3}\\
&= \frac{-8a^3}{12 t^3}\\
&= \frac{-2a^3}{3 t^3}\\
\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {2}}} &=- 3 \left(\frac{(1+i\sqrt{3})a}{\sqrt[3]{12} \cdot t} \right)^2 \left(\frac{(1-i\sqrt{3})t}{2 \sqrt[3]{18}} \right)\\
&= \frac{3a(1+2\sqrt{3}i-3)(1-i\sqrt{3})}{t \cdot 2 \cdot 2 \sqrt[3]{3 \cdot 3 \cdot 2 \cdot 18}}\\
&= \frac{3a(1+2\sqrt{3}i - 3- i\sqrt{3}+2\cdot 3 + i\sqrt[3]{3})}{4t \cdot 3 \sqrt{6}}\\
&= \frac{a(1-3+4\sqrt{3}i + 6)}{4t\sqrt[3]{6}}\\
&= \frac{a(4+4\sqrt{3}i)}{4t \sqrt[3]{6}}\\
&= \frac{a(1+\sqrt{3}i)}{t \sqrt[3]{6}}\\
\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {3}}} &= \frac{3(1+i\sqrt{3})a (1-2i\sqrt{3} - 3)t}{\sqrt[3]{12 \cdot 18^2}}\\
&= \frac{3at((1-2i\sqrt{3}-3)+(i\sqrt{3} + 2\cdot 3 - 3i\sqrt{3}))}{\sqrt[3]{2^2 \cdot 3 \cdot (2 \cdot 3^2)^2}}\\
&=
\end{align}
\textbf{Case 2.3:}
\todo[inline]{calculate...}
\[x = \frac{(1-i \sqrt{3})a}{\sqrt[3]{12} \cdot t} \[x = \frac{(1-i \sqrt{3})a}{\sqrt[3]{12} \cdot t}
-\frac{(1+i\sqrt{3}) t}{2\sqrt[3]{18}}\] -\frac{(1+i\sqrt{3}) t}{2\sqrt[3]{18}}\]