NORMALIZED CUBIC EQUATION ROOTS

NORMALIZED CUBIC EQUATION ROOTS    .

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Normalized Cubic Equation The single real root is blue Analytical expressions for the curves are developed at cubic_exp.xml from the expressions reported at Weisstein   wikipedia and by   Knaust   The real and imaginary components of the derived expressions were found to be exact fits.

Exact agreement for the first real root when all roots are real. $z=\sqrt{\frac{n}{2}}\bullet \sqrt{\frac{4\bullet {m}^{3}}{{n}^{2}}}\bullet 2\bullet \mathrm{cos}\left(\frac{0\bullet \pi }{6}+\frac{\mathrm{arctan}\left(\sqrt{\frac{4\bullet {m}^{3}}{{n}^{2}}-1}\right)}{3}\right)-\frac{1}{3}$

The imaginary components of the complex roots are red color coded curves.

The real component of the complex roots is color coded orange.

The real component of the complex roots is color coded orange.

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green $z=\sqrt{\frac{n}{2}}\bullet \sqrt{\frac{4\bullet {m}^{3}}{{n}^{2}}}\bullet \mathrm{cos}\left(\frac{0\bullet \pi }{6}+\frac{\mathrm{arctan}\left(\sqrt{\frac{4\bullet {m}^{3}}{{n}^{2}}-1}\right)}{3}\right)-\frac{1}{3}±\sqrt{\frac{n}{2}}\bullet \sqrt{\frac{4\bullet {m}^{3}}{{n}^{2}}}\bullet \sqrt{3}\bullet \mathrm{sin}\left(\frac{0\bullet \pi }{6}+\frac{\mathrm{arctan}\left(\sqrt{\frac{4\bullet {m}^{3}}{{n}^{2}}-1}\right)}{3}\right)$

maroon For an angle of pi/7 the solution is transcendental but for pi/60 the solution is only irrational

$2\bullet \mathrm{cos}\left(2\bullet \pi /7\right)={\left(\frac{7}{\mathrm{27}}\bullet \frac{\sqrt{1+{3}^{3}}}{2}\right)}^{\left(1/3\right)}2\bullet \mathrm{sin}\left(\frac{\pi -\frac{2}{3}\mathrm{atan}\left(\sqrt{\mathrm{27}}\right)}{2}\right)-\frac{1}{3}$

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