$\LaTeX$ Tests
Here are some tests of $\LaTeX$ using MathJax.
$$\int_{-\infty}^{\infty} e^{-x^2} dx$$Is written with two dollar signs: $$\int_{-\infty}^{\infty} e^{-x^2} dx$$
inline example: $\sum_{i = 0}^N 2i = y$
is made with a single dollar sign:
$\sum_{i = 0}^N 2i = y$
One overbrace:
$${a}^{b} - \overbrace{c}^{d}$$Like so: $${a}^{b} - \overbrace{c}^{d}$$
Like this: $$\underbrace{a}_{b} - \underbrace{c}_{d}$$
Testing defining new macros in Hugo/MathJax
$$ \RR \dd{x} $$Some arrays and matrices:
$$ \begin{aligned} \mathrm{equation1} &= 16 \\ \mathrm{equation2} &= 26 + 13 \end{aligned} $$$$ \begin{bmatrix} a & b \\ c & d \end{bmatrix} $$are written like this:
$$
\begin{aligned}
\mathrm{equation1} &= 16 \\
\mathrm{equation2} &= 26 + 13
\end{aligned}
$$
$$
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
$$
Fancy Letters
$$ \begin{aligned} \mathcal{ABCDEFGHIJKLMNOPQRSTUVWXYZ} & & \mathrm{\\mathcal\{ABC...\}} \\ \end{aligned} $$Some Famous Equations
Sturm-Liouville Operator
$$ \mathcal{L}[u] = \frac{d}{dx}\left[p(x)\frac{d}{dx}u\right] + q(x)\,u + \lambda w(x)\,u = 0 $$The Sturm-Liouville problem has some cool provable properties. And any second order ordinary differential equation can be cast into S-L form. Some of those properties are:
- infinite number of real, unique eigenvalues $\lambda_1, \lambda_2, \lambda_3, …$
- Corresponding to each $\lambda_n$ is an eigenfunction $u_n$ with exactly $n - 1$ zeros in $[a, b]$
- Eigenfunctions are orthogonal with weight $w$
- Eigenfunctions form a complete basis for a Hilbert space $L^2$ of functions
on some finite interval $[a, b]$ and where $\delta_{nm}$ is the Kronecker delta function.
- $p, q, w$ and $p’$ are continuous on $[a, b]$
- $p(x) > 0$ and $w(x) > 0$ for all $x \in [a,,b]$