# MathML polyfill

## Overview

This polyfill loads MathJax when MathML is detected on the page and the browser has inadequate MathML support.

### Known issues

• Browsers that lack MathML support (e.g. Chromium and IE11) will take longer to load MathML than browsers with MathML support
• IE11 won't display MathML correctly in isolated networks. It relies on an extra polyfill that requires internet access.

## Examples of simple formulas

Given the quadratic equation $a{x}^{2}+bx+c=0$ , the roots are given by $x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}$ .

## Examples of complex formulas

Formula Result
Bernoulli Trials $P\left(E\right)\text{Probability of event E: Get exactly k heads in n coin flips.}=\left(\genfrac{}{}{0}{}{n}{k}\right)\text{Number of ways to get exactly k heads in n coin flips}{p\text{Probability of getting heads in one flip}}_{}^{k\text{Number of heads}}{\left(1-p\right)\text{Probability of getting tails in one flip}}_{}^{n-k\text{Number of tails}}$
Cauchy-Schwarz Inequality ${\left(\sum _{k=1}^{n}{a}_{k}^{}{b}_{k}^{}\right)}_{}^{2}\le \left(\sum _{k=1}^{n}{a}_{k}^{2}\right)\left(\sum _{k=1}^{n}{b}_{k}^{2}\right)$
Cauchy Formula $f\left(z\right)\text{\hspace{0.17em}}·{\mathrm{Ind}}_{\gamma }^{}\left(z\right)=\frac{1}{2\pi i}\underset{\gamma }{\overset{}{\oint }}\frac{f\left(\xi \right)}{\xi -z}\text{\hspace{0.17em}}d\xi$
Cross Product ${V}_{1}^{}×{V}_{2}^{}=|\begin{array}{ccc}i& j& k\\ \frac{\partial X}{\partial u}& \frac{\partial Y}{\partial u}& 0\\ \frac{\partial X}{\partial v}& \frac{\partial Y}{\partial v}& 0\end{array}|$
Vandermonde Determinant $|\begin{array}{cccc}1& 1& \cdots & 1\\ {v}_{1}^{}& {v}_{2}^{}& \cdots & {v}_{n}^{}\\ {v}_{1}^{2}& {v}_{2}^{2}& \cdots & {v}_{n}^{2}\\ ⋮& ⋮& \ddots & ⋮\\ {v}_{1}^{n-1}& {v}_{2}^{n-1}& \cdots & {v}_{n}^{n-1}\end{array}|=\prod _{1\le i
Lorenz Equations $\begin{array}{rcl}\underset{}{\overset{˙}{x}}& =& \sigma \left(y-x\right)\\ \underset{}{\overset{˙}{y}}& =& \rho x-y-xz\\ \underset{}{\overset{˙}{z}}& =& -\beta z+xy\end{array}$
Maxwell's Equations $\left\{\begin{array}{rcl}\nabla \text{​}×\underset{}{\overset{↼}{B}}-\text{\hspace{0.17em}}\frac{1}{c}\text{\hspace{0.17em}}\frac{\partial \text{​}\underset{}{\overset{↼}{E}}}{\partial \text{​}t}& =& \frac{4\pi }{c}\text{\hspace{0.17em}}\underset{}{\overset{↼}{j}}\\ \nabla \text{​}·\underset{}{\overset{↼}{E}}& =& 4\pi \rho \\ \nabla \text{​}×\underset{}{\overset{↼}{E}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{1}{c}\text{\hspace{0.17em}}\frac{\partial \text{​}\underset{}{\overset{↼}{B}}}{\partial \text{​}t}& =& \underset{}{\overset{↼}{0}}\\ \nabla \text{​}·\underset{}{\overset{↼}{B}}& =& 0\end{array}$
Einstein Field Equations ${R}_{\mu \nu }^{}-\frac{1}{2}\text{\hspace{0.17em}}{g}_{\mu \nu }^{}\text{\hspace{0.17em}}R=\frac{8\pi G}{{c}_{}^{4}}\text{\hspace{0.17em}}{T}_{\mu \nu }^{}$
Ramanujan Identity $\frac{1}{\left(\sqrt{\phi \sqrt{5}}-\phi \right){e}_{}^{\frac{25}{\pi }}}=1+\frac{{e}_{}^{-2\pi }}{1+\frac{{e}_{}^{-4\pi }}{1+\frac{{e}_{}^{-6\pi }}{1+\frac{{e}_{}^{-8\pi }}{1+\dots }}}}$
Another Ramanujan identity $\sum _{k=1}^{\infty }\frac{1}{{2}_{}^{⌊k·\text{​}\phi ⌋}}=\frac{1}{{2}_{}^{0}+\frac{1}{{2}_{}^{1}+\cdots \frac{1}{{2}_{}^{1}+\frac{1}{{2}_{}^{2}+\cdots \frac{1}{{2}_{}^{3}+\frac{1}{{2}_{}^{5}+\dots }}}}}}$
Rogers-Ramanujan Identity
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