User:Zelaron

$$f(x,t) = \frac{6 i \sqrt{3} \left(\sqrt{t^3 \left(t (p t+r)^2-4 (q t+3)^3\right)}-p t^3-9 s t^2\right)}{-2 t^4 \left(p^2-4 q^3\right)+2 t^3 \left(36 q^2-p (r+9 s)\right)+18 s \sqrt{t^3 \left(t (p t+r)^2-4 (q t+3)^3\right)}+2 p t \sqrt{t^3 \left(t (p t+r)^2-4 (q t+3)^3\right)}+t^2 \left(216 q-r^2-81 s^2\right)+216 t}$$

$$\begin{vmatrix} -6 a_0 & a_1 & -a_2 & 0 \\ a_1 & -\frac{4 a_2}{9} & a_3 & 0 \\ 0 & a_3 & -9 a_4 & 9 a_0 \\ a_3 & -\frac{8 a_4}{9} & 0 & a_1 \end{vmatrix}$$

$$f\left(\frac{a+b\pm\sqrt{m(2-m)(a-b)^2}}{2}\right) = \frac{f(a)+f(b)\pm\sqrt{m(2-m)(f(a)-f(b))^2}}{2}$$

$$f\left(\frac{1}{3}\left(a+b+c\pm\sqrt{a^2+b^2+c^2-a b-a c-b c}\right)\right) = \frac{1}{3} \left(f(a) + f(b) + f(c) \pm \sqrt{f(a)^2 + f(b)^2 + f(c)^2 - f(a) f(b) - f(a) f(c) - f(b) f(c)}\right)$$

$$(\nu d+1)P(u) + \nu(z-u)P'(u)$$

$$\color{White}p(3) = 1 - \left(\frac{3}{(2\pi)^3}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\frac{\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z}{3-\cos{x}-\cos{y}-\cos{z}}\right)^{-1}\approx 0.340573$$

$$Q_2(u) = \frac{\mathfrak{d}^2 (\alpha - \mathfrak{d}) \left(3 \mathfrak{d}^2 a_{\mathfrak{d}-3} (-\alpha +\mathfrak{d}-3) (\mathfrak{d}-\alpha )^2+(\mathfrak{d}-1) \left(a_{\mathfrak{d}-1} \left(a_{\mathfrak{d}-1} (-\alpha +\mathfrak{d}-1) \left((\mathfrak{d}-1) a_{\mathfrak{d}-1} (\alpha  -\mathfrak{d}+1)^2+\mathfrak{d} \left(\mathfrak{d}^2 u-\mathfrak{d} (2 \alpha  u+u-2 z)+\alpha  (\alpha  u+u+z)-3 z\right)\right)-\mathfrak{d}^2 z (2 \mathfrak{d} (z-u)+2 \alpha  u+(\alpha -3) z)\right)+\mathfrak{d}^3 z^2 (\mathfrak{d}   (u-z)-\alpha  u+z)\right)+\mathfrak{d} a_{\mathfrak{d}-2} (\mathfrak{d}-\alpha ) \left(-(3 \mathfrak{d}-4) a_{\mathfrak{d}-1} (-\alpha +\mathfrak{d}-2) (-\alpha +\mathfrak{d}-1)-2 \mathfrak{d} \left(\alpha  (-2 \mathfrak{d} u+2 u+z)+(\mathfrak{d}-2) (\mathfrak{d} u+2 z)+\alpha ^2 u\right)\right)\right)}{\left((\mathfrak{d}-1) \left(a_{\mathfrak{d}-1}^2 (\alpha -\mathfrak{d}+1)^2+2 \mathfrak{d} z a_{\mathfrak{d}-1}+\mathfrak{d}^2 z^2\right)-2 \mathfrak{d} a_{\mathfrak{d}-2} (\alpha -\mathfrak{d}) (\alpha -\mathfrak{d}+2)\right){}^2}.$$

$$\mathrm{Discriminant}(F(u),u) = e^{i \pi \mathfrak{d} (3 \mathfrak{d}-7)/2}\,\mathfrak{d}^2(\alpha - \mathfrak{d}) \left(\alpha a_0+a_1 z+\frac{\left((\alpha -1) a_1+(\mathfrak{d}-1) \mathfrak{d} z\right) \left((\alpha - \mathfrak{d}+1)a_{\mathfrak{d}-1} + \mathfrak{d} z\right)}{\mathfrak{d}^2 (\mathfrak{d}-\alpha )}\right)$$

$$\mathrm{Discriminant}(F(u),u) = e^{i \pi \mathfrak{d} (5 \mathfrak{d}-13)/2}\, (\alpha -\mathfrak{d}) \frac{1}{\mathfrak{d}^2}\left(\frac{(\mathfrak{d}-1) \left(a_{\mathfrak{d}-1}^2 (\alpha -\mathfrak{d}+1)^2+2 \mathfrak{d} z a_{\mathfrak{d}-1}+\mathfrak{d}^2 z^2\right)}{\mathfrak{d}-\alpha }+2 \mathfrak{d}   a_{\mathfrak{d}-2} (\alpha -\mathfrak{d}+2)\right)^2 \left(\color{Green}(\alpha-1)a_1+2 a_2 z\color{Black}-\frac{\left(\color{Red}\alpha  a_0+a_1 z+\dfrac{\left((\alpha -1) a_1+2 a_2 z\right) \left((\alpha -\mathfrak{d}+1)a_{\mathfrak{d}-1}+\mathfrak{d} z\right)}{\mathfrak{d}^2 (\mathfrak{d}-\alpha )}\color{Black}\right)\mathfrak{d}^2 (\alpha -\mathfrak{d}) \left(3 \mathfrak{d}^2 a_{\mathfrak{d}-3} (-\alpha +\mathfrak{d}-3) (\mathfrak{d}-\alpha )^2+(\mathfrak{d}-1) \left(a_{\mathfrak{d}-1} \left(a_{\mathfrak{d}-1} (-\alpha   +\mathfrak{d}-1) \left((\mathfrak{d}-1) a_{\mathfrak{d}-1} (\alpha -\mathfrak{d}+1)^2+\mathfrak{d} z (\alpha +2 \mathfrak{d}-3)\right)-\mathfrak{d}^2 z^2 (\alpha +2 \mathfrak{d}-3)\right)+(1-\mathfrak{d}) \mathfrak{d}^3 z^3\right)+\mathfrak{d} a_{\mathfrak{d}-2} (\mathfrak{d}-\alpha ) \left(-(3 \mathfrak{d}-4) a_{\mathfrak{d}-1} (-\alpha +\mathfrak{d}-2) (-\alpha +\mathfrak{d}-1)-2 \mathfrak{d} z (\alpha +2 \mathfrak{d}-4)\right)\right)}{\left((\mathfrak{d}-1) \left(a_{\mathfrak{d}-1}^2 (\alpha -\mathfrak{d}+1)^2+2 \mathfrak{d} z  a_{\mathfrak{d}-1}+\mathfrak{d}^2 z^2\right)-2 \mathfrak{d} a_{\mathfrak{d}-2} (\alpha -\mathfrak{d}) (\alpha -\mathfrak{d}+2)\right)^2}\right)$$

$$\lim_{D_1\to 33/100,\,D_2\to 83/50}\left| \begin{array}{cccccc} \frac{6}{\Gamma \left(4-D_1\right)} & \frac{2 a}{\Gamma \left(3-D_1\right)} & \frac{b}{\Gamma \left(2-D_1\right)} & \frac{c}{\Gamma \left(1-D_1\right)} & 0 & 0 \\ 0 & \frac{6}{\Gamma \left(4-D_1\right)} & \frac{2 a}{\Gamma \left(3-D_1\right)} & \frac{b}{\Gamma \left(2-D_1\right)} & \frac{c}{\Gamma \left(1-D_1\right)} & 0 \\ 0 & 0 & \frac{6}{\Gamma \left(4-D_1\right)} & \frac{2 a}{\Gamma \left(3-D_1\right)} & \frac{b}{\Gamma \left(2-D_1\right)} & \frac{c}{\Gamma \left(1-D_1\right)} \\ \frac{6}{\Gamma \left(4-D_2\right)} & \frac{2 a}{\Gamma \left(3-D_2\right)} & \frac{b}{\Gamma \left(2-D_2\right)} & \frac{c}{\Gamma \left(1-D_2\right)} & 0 & 0 \\ 0 & \frac{6}{\Gamma \left(4-D_2\right)} & \frac{2 a}{\Gamma \left(3-D_2\right)} & \frac{b}{\Gamma \left(2-D_2\right)} & \frac{c}{\Gamma \left(1-D_2\right)} & 0 \\ 0 & 0 & \frac{6}{\Gamma \left(4-D_2\right)} & \frac{2 a}{\Gamma \left(3-D_2\right)} & \frac{b}{\Gamma \left(2-D_2\right)} & \frac{c}{\Gamma \left(1-D_2\right)} \\ \end{array} \right|\approx a^2 b^2 - 3.79847 b^3 - 3.59317 a^3 c + 15.2455 a b c - 18.7119 c^2.$$

$$\left\vert \frac{u + u^2 + \sqrt{-3(u(u-1))^2}}{2 + 2u(u-1)}\right\vert \vert u - 1\vert = \left\vert \frac{u + u^2 + \sqrt{-3(u(u-1))^2}}{2 + 2u(u-1)}-1\right\vert \vert u\vert$$

$$\left\vert u^3-\sqrt{3} \sqrt{-(u-1)^2 u^2} u+\sqrt{3} \sqrt{-(u-1)^2 u^2}-u\right\vert = \left\vert -u^3+3 u^2-\sqrt{3} \sqrt{-(u-1)^2 u^2} u-2 u\right\vert$$

$$\left| z^3-\sqrt{-3(z-1)^2 z^2} z-z+\sqrt{-3(z-1)^2 z^2}\right| = \left| -z^3+3 z^2-\sqrt{-3(z-1)^2 z^2} z-2 z\right|$$

$$\left| (z-1) \left(z (z+1)-\sqrt{-3(z-1)^2 z^2}\right)\right| = \left| z \left((z-1)(z-2)+\sqrt{-3(z-1)^2 z^2}\right)\right|$$