AH Formatter MathML Demonstration

Elementary Math #1

$$123\times 321123246369$$

Elementary Math #2

$$325436.313061210916151.091$$

Simultaneous Equations

$$\begin{array}{c}8.44x+55y=0\\ 3.1x-0.7y=-1.1\end{array}$$

Quadratic Formula

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

Standard Deviation

$$\sigma =\sqrt{\frac{1}{N}\sum _{i=1}^N{(x_i-\mu )}^2}$$

Divergence

$$\nabla ·\overrightarrow{v}=\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+\frac{\partial v_z}{\partial z}$$

Complex Number

$$c=\stackrel{\text{complex number}}{\overbrace{\underset{\text{real}}{\underbrace{a}}+\underset{\text{imaginary}}{\underbrace{bi}}}}$$

De Morgan's Laws

$$\overline{\bigcup _{i\in I}{A}_i}\equiv \bigcap _{i\in \mathrm{I}}\overline{A_i}$$

Binomial Coefficient

$$C\left(n,k\right)=C_k^n={}_{n}C_k=\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!\left(n-k\right)!}$$

Bernoulli Trials

$$P(k)\text{Probability of event E: Get exactly k heads in n coin flips.}=\left(\genfrac{}{}{0ex}{}{n}{k}\right)\text{Number of ways to get exactly k heads in n coin flips}{p\text{Probability of getting heads in one flip}}_{}^{k\text{Number of heads}}{(1-p)\text{Probability of getting tails in one flip}}_{}^{n-k\text{Number of tails}}$$

Normal Distribution

$$f\left(x,\mu ,\sigma \right)=\frac{e^{-\frac{{\left(x-\mu \right)}^2}{2{\sigma }^2}}}{\sigma \sqrt{2\mathrm{\pi }}}$$

Lorenz Equations

$$\begin{array}{rcl}\dot{x}& =& \sigma (y-x)\\ \dot{y}& =& \rho x-y-xz\\ \dot{z}& =& -\beta z+xy\end{array}$$

Cross Product

$$V_1\times V_2=\left|\begin{array}{ccc}i& j& k\\ \frac{\partial X}{\partial u}& \frac{\partial Y}{\partial u}& 0\\ \frac{\partial X}{\partial v}& \frac{\partial Y}{\partial v}& 0\end{array}\right|$$

Axiom of Power Set

$$\forall A\exists P\forall B\left[B\in P⟺\forall C\left(C\in B\Rightarrow C\in A\right)\right]$$

Cauchy's Integral Formula

$$f\left(a\right)=\frac{1}{2\mathrm{\pi }i}{\oint }_{\gamma }\frac{f\left(z\right)}{z-a}dz$$

Einstein's Field Equations

$$R_{\mu \nu }-\frac{1}{2}{ɡ}_{\mu \nu }R+{ɡ}_{\mu \nu }\mathrm{\Lambda }=\frac{8\mathrm{\pi }G}{c^4}{T}_{\mu \nu }$$

Legendre's Differential Equation

$$\frac{d}{dx}\left[\left(1-x^2\right)\frac{d}{dx}{P}_n\left(x\right)\right]+n\left(n+1\right)P_n\left(x\right)=0$$

Sophomore's Dream

$${\int }_0^1{x}^{x}dx=\sum _{n=1}^{\mathrm{\infty }}{\left(-1\right)}^{n+1}{n}^{-n}=-\sum _{n=1}^{\mathrm{\infty }}{\left(-n\right)}^{-n}$$

Maxwell's Equations

$$\{\begin{array}{rcl}\nabla \times \stackrel{\leftharpoonup }{B}-\frac{1}{c}\frac{\partial \stackrel{\leftharpoonup }{E}}{\partial t}& =& \frac{4\mathrm{\pi }}{c}\stackrel{\leftharpoonup }{j}\\ \nabla ·\stackrel{\leftharpoonup }{E}& =& 4\mathrm{\pi }\rho \\ \nabla \times \stackrel{\leftharpoonup }{E}+\frac{1}{c}\frac{\partial \stackrel{\leftharpoonup }{B}}{\partial t}& =& \stackrel{\leftharpoonup }{0}\\ \nabla ·\stackrel{\leftharpoonup }{B}& =& 0\end{array}$$

Schwinger-Dyson Equation

$$⟨\psi \left|𝒯\left\{\frac{\delta }{\delta \varphi }F\left[\varphi \right]\right\}\right|\psi ⟩=-i⟨\psi \left|𝒯\left\{F\left[\varphi \right]\frac{\delta }{\delta \varphi }S\left[\varphi \right]\right\}\right|\psi ⟩$$

Ramanujan Identity

$$\frac{1}{(\sqrt{\varphi \sqrt{5}}-\varphi )e^{\frac{25}{\mathrm{\pi }}}}=1+\frac{e^{-2\mathrm{\pi }}}{1+\frac{e^{-4\mathrm{\pi }}}{1+\frac{e^{-6\mathrm{\pi }}}{1+\frac{e^{-8\mathrm{\pi }}}{1+\cdots }}}}$$

Rogers-Ramanujan Identity

$$\begin{array}{llll}\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n^2}}{{(q)}_{\mathrm{\infty }}}& =& \frac{1}{\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{5n-4}\right)\left(1-q^{5n-1}\right)}& ,\text{for }\left|q\right|<1\\ & =& 1+q+q^2+q^3+2q^4+2q^5+3q^6+\cdots \end{array}$$

Moore Determinant

$$M=\left[\begin{array}{cccc}{\alpha }_1& {\alpha }_1^q& \dots & {\alpha }_1^{q^{n-1}}\\ {\alpha }_2& {\alpha }_2^q& \dots & {\alpha }_2^{q^{n-1}}\\ ⋮& ⋮& \ddots & ⋮\\ {\alpha }_m& {\alpha }_m^q& \dots & {\alpha }_m^{q^{n-1}}\end{array}\right]$$

Differentiable Manifold

$${\gamma }_1\equiv {\gamma }_2⟺\left\{\begin{array}{l}{\gamma }_1\left(0\right)={\gamma }_2\left(0\right)=p\text{, and}\\ {\left.\frac{d}{dt}\varphi \circ {\gamma }_1\left(t\right)\right|}_{t=0}={\left.\frac{d}{dt}\varphi \circ {\gamma }_2\left(t\right)\right|}_{t=0}\end{array}\right.$$

Sphere Volume

Spherical coordinates derivation of the volume of a sphere $\left(\frac{4}{3}\pi R^3\right)$. The formula $S$ for a sphere of radius $R$ in spherical coordinates is: $S=\left\{0\le \varphi \le 2\pi ,0\le \theta \le \pi ,0\le \rho \le R\right\}$

$$\begin{array}{rl}\text{Volume}& =\underset{S}{\iiint }{\rho }^{2}sin\theta d\rho d\theta d\varphi \\ & ={\int }_0^{2\pi }d\varphi {\int }_0^{\pi }sin\theta d\theta {\int }_0^R{\rho }^{2}d\rho \\ & =\varphi {|}_0^{2\pi }\left(-\cos \theta \right){|}_0^{\pi }\frac{1}{3}{\rho }^3{|}_0^R\\ & =2\pi \times 2\times \frac{1}{3}{R}^3\\ & =\frac{4}{3}\pi R^3\end{array}$$

Cichoń's Diagram

$$\begin{array}{ccccccccccc}\multicolumn{2}{c}{}& cov\left(ℒ\right)& ⟶& non\left(𝒦\right)& ⟶& cof\left(𝒦\right)& ⟶& cof\left(ℒ\right)& ⟶& 2^{{\aleph }_0}\\ \uparrow & & \uparrow & & \uparrow & & \uparrow & \multicolumn{2}{c}{}\\ & 𝔟& ⟶& 𝔡\\ & \uparrow & & \uparrow \\ {\aleph }_1& ⟶& add\left(ℒ\right)& ⟶& add\left(𝒦\right)& ⟶& cov\left(𝒦\right)& ⟶& non\left(ℒ\right)\end{array}$$

Multiscripts & Greek Alphabet

$$\underset{{}_{\varphi }{}^{\chi }𝔉_{\omega }^{\psi }}{\overset{{}_{\iota }{}^{\kappa }ℭ_{\mu }^{\lambda }}{{}_{{}_{\alpha }{}^{\beta }𝔄_{\delta }^{\gamma }}{}^{{}_{\epsilon }{}^{\zeta }𝔅_{\theta }^{\eta }}\prod _{{}_{\rho }{}^{\sigma }𝔈_{\upsilon }^{\tau }}^{{}_{\nu }{}^{\xi }𝔇_{\pi }^o}}}$$

Stacked Exponents

$$g\left(z\right)=e^{-\sum _{i=0}^{\infty }{z}^{\frac{2}{a-i}}}$$

Nested Roots

$$\frac{\sqrt{1+\sqrt[3]{2+\sqrt[5]{3+\sqrt[7]{4+\sqrt[11]{5+\sqrt[13]{6+\sqrt[17]{7+\sqrt[19]{A}}}}}}}}}{e^{\pi }}=x^{‴}$$

Nested Matrices

$$\left(\begin{array}{cc}\left(\begin{array}{cccc}{a}_1& a_2& a_3& a_4\\ a_5& a_6& a_7& a_8\end{array}\right)& \left(\begin{array}{c}{b}_1\\ b_2\\ b_3\\ b_4\end{array}\right)\\ \begin{array}{cc}0\hspace{0.33em}& \left(\begin{array}{cc}{c}_1& c_2\\ c_3& c_4\end{array}\right)\end{array}\end{array}\right)$$

Font Sizes

$$\text{scriptlevel}:-3,-2,-1,0,1,2,3$$