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added calculation

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Martin Thoma 2013-12-19 13:37:25 +01:00
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commit 2a95674c76
2 changed files with 9 additions and 2 deletions
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documents/math-minimal-distance-to-cubic-function/quadratic-functions.tex
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@ -231,9 +231,16 @@ Now simplify the summands:
&= \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}}\\
\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {3}}} &= 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\\
&= \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}}\\
&=
&= \frac{3at(4+i(-4\sqrt{3})}{4\cdot3 \sqrt[3]{9}}\\
&= \frac{at(1-\sqrt{3}i}{\sqrt[3]{9}}\\
\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {4}}} &= - \left(\frac{(1-i\sqrt{3})t}{2 \sqrt[3]{18}}\right)^3\\
&= -\frac{(1-3i\sqrt{3} - 3 \cdot 3 + i \sqrt{27}) t^3}{8 \cdot 18}\\
&=- \frac{t^3 (-8)}{2^4 \cdot 3^2}\\
&= \frac{8t^3}{2^4 \cdot 3^2}\\
&= \frac{t^3}{18}
\end{align}