# Math Examples

#### CSS3 styling of math

${C}^{\infty }\text{-fns}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}e=m{c}^{2}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}{e}^{\pi i}=-1\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}w={x}^{\left({y}^{z}\right)}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}\frac{{x}^{2}}{2{x}^{3}-4x+1}$ ${x}_{{m}_{k}}={{x}_{{n}_{j}}}^{1⁄2}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}\sqrt[3]{\frac{\alpha \xi +\beta }{\gamma \xi +\delta }}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}{n}^{2}\equiv 1\phantom{\rule{0.2em}{0ex}}\left(mod4\right)\phantom{\rule{0.5em}{0ex}}\text{if}\phantom{\rule{0.5em}{0ex}}n\equiv ±1\phantom{\rule{0.2em}{0ex}}\left(mod2\right)\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}\Delta ABC\cong \Delta DEF$ $r=||x||=\sqrt{{{x}_{1}}^{2}+{{x}_{2}}^{2}+\dots +{{x}_{n}}^{2}}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}\frac{1+\sqrt{5}}{2}=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\dots }}}=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots }}}}}$ ${x}^{2}{y}^{2}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}M↦{}^{t}M^{-1}\phantom{\rule{0.5em}{0ex}}\text{has order}\phantom{\rule{0.5em}{0ex}}2\phantom{\rule{0.5em}{0ex}}\text{in}\phantom{\rule{0.5em}{0ex}}{\mathrm{GL}}_{n}\left(\mathbf{R}\right)\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}{}_{2}F_{3}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}z=x+{y}^{\left(\frac{2}{k+1}\right)}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}\frac{a}{b⁄2}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}\left(\genfrac{}{}{0}{}{n}{k⁄2}\right)$ ${\left(\frac{a}{b}\right)}^{\frac{1}{2}}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}\sqrt{\frac{a}{b}}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}\sqrt{\frac{\frac{a}{b}}{\frac{c}{d}}}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}{e}^{t}=\sum _{k=0}^{\infty }\phantom{\rule{0.2em}{0ex}}\frac{{t}^{k}}{k!}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}\phantom{\rule{0.1em}{0ex}}\mathrm{sin}ax\phantom{\rule{0.1em}{0ex}}\mathrm{cos}bx\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}\int {\int }_{S}\phantom{\rule{0.2em}{0ex}}\left(\mathbf{curl}\phantom{\rule{0.2em}{0ex}}\mathbf{F}\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}\mathbf{N}\right)\phantom{\rule{0.2em}{0ex}}d\sigma ={\int }_{\partial S}\phantom{\rule{0.2em}{0ex}}\left(\mathbf{F}\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}\mathbf{T}\right)\phantom{\rule{0.2em}{0ex}}ds$ ${\left(1+t\right)}^{r}=\sum _{k=0}^{\infty }\phantom{\rule{0.2em}{0ex}}\frac{r\left(r-1\right)\left(r-2\right)\dots \left(r-k+1\right)}{k!}\phantom{\rule{0.2em}{0ex}}{t}^{k}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}{D}^{2}y-3x{\left(Dy\right)}^{2}=x\phantom{\rule{0.1em}{0ex}}\mathrm{cos}x$ $\left(\frac{{\partial }^{2}}{\partial {x}^{2}}+\frac{{\partial }^{2}}{\partial {y}^{2}}\right){\left|\phi \left(x+iy\right)\right|}^{2}=0\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}{}_{b}{}^{a}\mathrm{Hom}_{c}^{d}\left(X,Y\right)\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}X\stackrel{f}{\to }Y\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}X\underset{f}{\to }Y\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}\stackrel{A}{X}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}\underset{A}{X}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}\mathrm{Gal}\left(\overline{\mathbf{Q}}⁄\mathbf{Q}\right)$ $\overline{X}\phantom{\rule{1.8em}{0ex}}\stackrel{˘}{X}\phantom{\rule{1.8em}{0ex}}\stackrel{ˇ}{X}\phantom{\rule{1.8em}{0ex}}\stackrel{˙}{X}\phantom{\rule{1.8em}{0ex}}\stackrel{¨}{X}\phantom{\rule{1.8em}{0ex}}\stackrel{^}{X}\phantom{\rule{1.8em}{0ex}}\stackrel{\sim }{X}\phantom{\rule{1.8em}{0ex}}\stackrel{⇀}{X}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}T_{{j}_{1}{j}_{2}\dots {j}_{q}}^{{i}_{1}{i}_{2}\dots {i}_{p}}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{1.8em}{0ex}}E_{2}^{pq}={H}^{p}\left(B,{H}^{q}\left(F\right)\right)⇒{H}^{*}\left(X\right)$ $\frac{1}{1+\frac{{e}^{-2\pi \sqrt{5}}}{1+\frac{{e}^{-4\pi \sqrt{5}}}{1+\frac{{e}^{-6\pi \sqrt{5}}}{\dots }}}}=\left(\frac{\sqrt{5}}{1+\sqrt[5]{{5}^{3⁄4}{\left(\frac{\sqrt{5}-1}{2}\right)}^{5⁄2}-1}}-\frac{\sqrt{5}+1}{2}\right){e}^{2\pi ⁄\sqrt{5}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\text{(Ramanujan)}$