# Math Rendering Test

## Inline Math

The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ and Euler's identity is $e^{i\pi} + 1 = 0$.

## Display Math

$$\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}$$

$$\sum_{n=0}^{\infty} \frac{x^n}{n!} = e^x$$

## Matrices

$$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}$$

## Aligned Equations

$$\begin{aligned}
\nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_0} \\
\nabla \cdot \mathbf{B} &= 0 \\
\nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\
\nabla \times \mathbf{B} &= \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}
\end{aligned}$$

## Fractions and Nested Expressions

$$\cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cdots}}}}$$

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

## Limits and Calculus

$$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e$$

$$\oint_{\partial \Sigma} \mathbf{B} \cdot d\boldsymbol{\ell} = \mu_0 \iint_{\Sigma} \mathbf{J} \cdot d\mathbf{S} + \mu_0 \varepsilon_0 \frac{d}{dt} \iint_{\Sigma} \mathbf{E} \cdot d\mathbf{S}$$

## Greek and Symbols

$$\Gamma(z) = \int_0^{\infty} t^{z-1} e^{-t} \, dt, \quad \Re(z) > 0$$

$$\mathcal{L}\{f(t)\} = \int_0^{\infty} f(t) e^{-st} \, dt$$

## Cases

$$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$