User:Zelaron

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

${\begin{vmatrix}-6a_{0}&a_{1}&-a_{2}&0\\a_{1}&-{\frac {4a_{2}}{9}}&a_{3}&0\\0&a_{3}&-9a_{4}&9a_{0}\\a_{3}&-{\frac {8a_{4}}{9}}&0&a_{1}\end{vmatrix}}$

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

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

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

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

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

$\mathrm {Discriminant} (F(u),u)=e^{i\pi {\mathfrak {d}}(3{\mathfrak {d}}-7)/2}\,{\mathfrak {d}}^{2}(\alpha -{\mathfrak {d}})\left(\alpha a_{0}+a_{1}z+{\frac {\left((\alpha -1)a_{1}+({\mathfrak {d}}-1){\mathfrak {d}}z\right)\left((\alpha -{\mathfrak {d}}+1)a_{{\mathfrak {d}}-1}+{\mathfrak {d}}z\right)}{{\mathfrak {d}}^{2}({\mathfrak {d}}-\alpha )}}\right)$

$\mathrm {Discriminant} (F(u),u)=e^{i\pi {\mathfrak {d}}(5{\mathfrak {d}}-13)/2}\,(\alpha -{\mathfrak {d}}){\frac {1}{{\mathfrak {d}}^{2}}}\left({\frac {({\mathfrak {d}}-1)\left(a_{{\mathfrak {d}}-1}^{2}(\alpha -{\mathfrak {d}}+1)^{2}+2{\mathfrak {d}}za_{{\mathfrak {d}}-1}+{\mathfrak {d}}^{2}z^{2}\right)}{{\mathfrak {d}}-\alpha }}+2{\mathfrak {d}}a_{{\mathfrak {d}}-2}(\alpha -{\mathfrak {d}}+2)\right)^{2}\left(\color {Green}(\alpha -1)a_{1}+2a_{2}z\color {Black}-{\frac {\left(\color {Red}\alpha a_{0}+a_{1}z+{\dfrac {\left((\alpha -1)a_{1}+2a_{2}z\right)\left((\alpha -{\mathfrak {d}}+1)a_{{\mathfrak {d}}-1}+{\mathfrak {d}}z\right)}{{\mathfrak {d}}^{2}({\mathfrak {d}}-\alpha )}}\color {Black}\right){\mathfrak {d}}^{2}(\alpha -{\mathfrak {d}})\left(3{\mathfrak {d}}^{2}a_{{\mathfrak {d}}-3}(-\alpha +{\mathfrak {d}}-3)({\mathfrak {d}}-\alpha )^{2}+({\mathfrak {d}}-1)\left(a_{{\mathfrak {d}}-1}\left(a_{{\mathfrak {d}}-1}(-\alpha +{\mathfrak {d}}-1)\left(({\mathfrak {d}}-1)a_{{\mathfrak {d}}-1}(\alpha -{\mathfrak {d}}+1)^{2}+{\mathfrak {d}}z(\alpha +2{\mathfrak {d}}-3)\right)-{\mathfrak {d}}^{2}z^{2}(\alpha +2{\mathfrak {d}}-3)\right)+(1-{\mathfrak {d}}){\mathfrak {d}}^{3}z^{3}\right)+{\mathfrak {d}}a_{{\mathfrak {d}}-2}({\mathfrak {d}}-\alpha )\left(-(3{\mathfrak {d}}-4)a_{{\mathfrak {d}}-1}(-\alpha +{\mathfrak {d}}-2)(-\alpha +{\mathfrak {d}}-1)-2{\mathfrak {d}}z(\alpha +2{\mathfrak {d}}-4)\right)\right)}{\left(({\mathfrak {d}}-1)\left(a_{{\mathfrak {d}}-1}^{2}(\alpha -{\mathfrak {d}}+1)^{2}+2{\mathfrak {d}}za_{{\mathfrak {d}}-1}+{\mathfrak {d}}^{2}z^{2}\right)-2{\mathfrak {d}}a_{{\mathfrak {d}}-2}(\alpha -{\mathfrak {d}})(\alpha -{\mathfrak {d}}+2)\right)^{2}}}\right)$

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

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

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

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

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