A KaTeX Rendering Tour

This is an evaluation post. Its only job is to show how the math rules in src/utils/markdown.js render to the page, so the KaTeX integration can be reviewed. Delete it when you are done.

Delimiters: inline math is $...$, display math is $$...$$ (single-line or multi-line). Two conventions: no space directly inside the delimiters, and a $ that immediately follows a word character is never treated as an opener. Red text means KaTeX refused to render the formula (see the last sections).

Inline math permalink

Euler's identity $e^{i\pi}+1=0$ is often called the most beautiful equation in mathematics. A single sentence can carry several formulas: the area of a circle is $A=\pi r^2$, the slope of $y=mx+b$ is $m$, and the discriminant $\Delta=b^2-4ac$ decides whether the roots of $ax^2+bx+c=0$ are real.

Sub- and superscripts chain: $x_i^{(2)}$ is the second iterate of the $i$-th component, while $f^{-1}(g^{-1}(x))$ composes inverse functions, and primes work as $f' = f'' = f'''$.

Inline big operators tuck their limits in: $\sum_{i=1}^{n} i = \tfrac{n(n+1)}{2}$ and $\lim_{x\to 0} \frac{\sin x}{x} = 1$. Force limits above and below with $\sum\limits_{i=1}^{n}$ and $\prod\limits_{k=0}^{\infty}$.

Prices stay literal — a 5$ resistor and a 10$ capacitor, or a $5 part and a $10 board: no math mode is entered, because the delimiters here never satisfy the opener rules.

Display blocks permalink

The quadratic formula, single-line block:

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

Multi-line blocks continue until a line ending in $$ — Maxwell's equations in differential form:

$$\begin{aligned} \nabla \times \mathbf{B} &= \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \\ \nabla \cdot \mathbf{E} &= \frac{\rho}{\epsilon_0} \\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\ \nabla \cdot \mathbf{B} &= 0 \end{aligned}$$

align lines up at the & positions — the Lorenz system:

$$\begin{align} x' &= \sigma (y - x) \\ y' &= x (\rho - z) - y \\ z' &= xy - \beta z \end{align}$$

And align*:

$$\begin{align*} a^2 + b^2 &= c^2 \\ \tan\theta &= \frac{b}{a} \end{align*}$$

equation wraps a single formula (it does not get an automatic number in this build — see "Numbering and tags"):

$$\begin{equation} e^{i\pi} + 1 = 0 \end{equation}$$

gathered centers every line:

$$\begin{gathered} (x+y+z)^2 = x^2 + y^2 + z^2 \\ + 2xy + 2xz + 2yz \end{gathered}$$

Matrices permalink

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

Round, square, and vertical bars; cases for piecewise functions:

$$\mathbf{u} = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}, \qquad \begin{Bmatrix} a & b \\ c & d \end{Bmatrix}, \qquad P(t) = \begin{cases} 0, & t < 0 \\ t, & 0 \le t \le 1 \\ 2 - t, & 1 < t \le 2 \\ 0, & t > 2 \end{cases}$$

Inline matrices stay at text scale: $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$, $\begin{smallmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{smallmatrix}$, and the Euclidean norm is $\lVert \mathbf{x} \rVert_2 = \sqrt{x_1^2 + \cdots + x_n^2}$.

Big operators and limits permalink

Display mode puts limits above and below:

$$\sum_{i=1}^{n} \prod_{j=1}^{m} x_{ij} \qquad \int_{0}^{\infty} e^{-x^2}\,dx = \frac{\sqrt{\pi}}{2} \qquad \iint_{D} f(x,y)\,dx\,dy \qquad \oint_{\gamma} f(z)\,dz \qquad \bigcup_{i=1}^{n} A_i$$

Limits that read as limits:

$$\lim_{x\to 0} \frac{e^x - 1}{x} = 1, \qquad \limsup_{n\to\infty} a_n \le \liminf_{n\to\infty} b_n, \qquad \lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^n = e$$

Fractions, radicals, continued fractions permalink

Nested fractions and indexed roots:

$$\frac{1}{1 + \frac{2}{3 + \frac{4}{5}}} \qquad \sqrt{2} + \sqrt[3]{2} + \sqrt[n]{\frac{a}{b}} \qquad \frac{\partial f}{\partial x} \bigg|_{x=0}$$

A continued fraction, the kind of thing that would be painful to lay out by hand:

$$\cfrac{1}{2 + \cfrac{1}{3 + \cfrac{1}{4 + \cfrac{1}{5}}}}$$

Accents, bars, and brackets permalink

$$\hat{\theta} \qquad \vec{v} \qquad \bar{x} \qquad \tilde{f} \qquad \widehat{xyz} \qquad \widetilde{xyz} \qquad \dot{x} \qquad \ddot{x}$$ $$\overline{AB} \qquad \underline{\sum} \qquad \overbrace{a + b + c}^{\text{the sum of the terms}} \qquad \underbrace{x \cdot x \cdot x}_{\text{the cube}} \qquad a \stackrel{\text{def}}{=} b \qquad \underset{\gamma}{\lim} \qquad \overset{+}{-}$$

Fonts and text permalink

$$\mathbf{x} \qquad \bm{A} \qquad \pmb{x} \qquad \mathit{x} \qquad \mathrm{d}x \qquad \mathcal{L} \qquad \mathbb{R} \qquad \mathfrak{g} \qquad \mathsf{x} \qquad \mathtt{x}$$

Text inside math: $\text{for } x \in \text{dom}(f)$, $\textbf{bold text}$, $\textit{italic text}$, and $\operatorname{erf}(x)$ for a roman operator.

Delimiters, sizing, and spacing permalink

Auto-sizing delimiters scale to their content; \big through \Bigg do it manually; \left. and \right| size invisible anchors:

$$\left( \frac{a}{b} \right) \qquad \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \qquad \left\{ \frac{a}{b} \right\} \qquad \left. \frac{d}{dx} \right|_{x=0} \qquad \bigg( \frac{a}{b} \bigg) \quad \Bigg( \frac{a}{b} \Bigg)$$

Custom fraction delimiters with \genfrac, and the binomial coefficient:

$$\genfrac{[}{]}{0pt}{}{a}{b} \qquad \binom{n}{k} = \frac{n!}{k!(n-k)!} \qquad {n \choose k}$$

Spacing: thin $a\,b$, thick $a\;b$, negative $a\!b$, word $a\quad b$, and double-word $a\qquad b$.

Symbols permalink

Relations: $a \le b$, $a \ge b$, $a \ne b$, $a \approx b$, $a \equiv b$, $a \sim b$, $a \simeq b$, $a \cong b$, $a \propto b$, $a \ll b$, $a \gg b$, $a \leqslant b$.

Operators: $a + b$, $a \pm b$, $a \mp b$, $a \times b$, $a \cdot b$, $a \circ b$, $a \bullet b$, $a \div b$, $a \bmod b$, $a \pmod{b}$, $a \mid b$, $a \nmid b$.

Arrows: $a \to b$, $a \mapsto b$, $a \rightarrow b$, $a \Rightarrow b$, $a \Leftarrow b$, $a \iff b$, $a \xrightarrow{\text{label}} b$, $a \hookrightarrow b$, $a \twoheadrightarrow b$.

Sets and logic: $a \in S$, $a \notin S$, $S \subset T$, $S \subseteq T$, $A \cup B$, $A \cap B$, $A \setminus B$, $\emptyset$, $\forall x$, $\exists y$, $\nexists z$, $\therefore$, $\because$.

Greek and geometry: $\alpha\,\beta\,\gamma\,\delta\,\varepsilon\,\zeta\,\eta\,\theta\,\vartheta\,\iota\,\kappa\,\lambda\,\mu\,\nu\,\xi\,\pi\,\rho\,\sigma\,\tau\,\upsilon\,\phi\,\varphi\,\chi\,\psi\,\omega$ and capitals $\Gamma\,\Delta\,\Theta\,\Lambda\,\Xi\,\Pi\,\Sigma\,\Phi\,\Psi\,\Omega$, plus $\angle ABC$, $\triangle ABC$, $a \perp b$, $a \parallel b$, $\hbar$, $\ell$, $\Re z$, $\Im z$, $\aleph_0$.

Color, boxes, and cancellation permalink

$$\textcolor{red}{\mathbf{x}} \qquad \textcolor{blue}{f(x)} \qquad \textcolor{#e67e22}{\frac{a}{b}} \qquad \colorbox{yellow}{\text{a colored box}} \qquad \boxed{\frac{-b \pm \sqrt{b^2-4ac}}{2a}}$$

Cancellation, in three flavors: $a \cancel{+} b$, $x \bcancel{-} y$, $z \xcancel{*} w$.

Numbering and tags permalink

This KaTeX build does not number equation or align lines automatically, and \label/\eqref are unsupported. Explicit numbering works with \tag:

$$e = \sum_{n=0}^{\infty} \frac{1}{n!} \tag{2.71828}$$

Local macros permalink

\newcommand works, but only inside the formula that defines it — macros do not leak across formulas:

$$\newcommand{\E}{\mathbb{E}} \qquad \newcommand{\Var}{\operatorname{Var}} \qquad \E[X] = \mu \qquad \Var(X) = \E[(X - \mu)^2]$$

Error handling permalink

KaTeX is called with throwOnError: false, so invalid math renders as readable red text at the failure point instead of breaking the build.

Unknown command:

$$\unknowncommand{x}$$

Unbalanced fraction:

\frac{1}

The \href command requires the trust option, which this site does not enable:

$$\href{https://example.org}{a link}$$

Known limitations permalink

• Math does not wrap. This inline formula overflows the column instead of breaking: $\det\begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \end{pmatrix} = \sum_{\sigma \in S_5} \operatorname{sgn}(\sigma) \prod_{i=1}^{5} a_{i,\sigma(i)}$
• No automatic equation numbering: equation and align render without numbers, and \label/\eqref are unsupported — use \tag for explicit numbers.
• \eqnarray, \multline, \widebar, \cancelto, \html, \href, \includegraphics, and \resizebox are unsupported in this KaTeX build (or disabled by default).
• Inline math is one line only; a $ pair spanning a newline is literal text.