Math Equations

Overview

KaTeX is a fast, easy-to-use JavaScript library for rendering TeX math notation on the web. This site integrates KaTeX to enable beautiful mathematical equations in documentation.

Features

Fast rendering - KaTeX is significantly faster than MathJax
High quality - Produces crisp output at any zoom level
Self-contained - No dependencies on external fonts or stylesheets
Server-side rendering - Works without JavaScript enabled
TeX/LaTeX syntax - Familiar notation for mathematicians

Basic Usage

Inline Math

Wrap inline equations with single dollar signs $...$ :

The Pythagorean theorem states that $c = \pm\sqrt{a^2 + b^2}$ for a right triangle.

The Pythagorean theorem states that $c = \pm\sqrt{a^2 + b^2}$ for a right triangle.

Block Math

Use code blocks with the math language identifier or wrap with double dollar signs $$...$$:

```math
c = \pm\sqrt{a^2 + b^2}
```

c = \pm\sqrt{a^2 + b^2}

Or using double dollar signs:

$$
E = mc^2
$$

E = mc^2

Common Examples

Algebra

Quadratic Formula:

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

Binomial Theorem:

(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k

Calculus

Fundamental Theorem of Calculus:

\int_a^b f(x) \, dx = F(b) - F(a)

Partial Derivatives:

\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}

Limit Definition:

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

Linear Algebra

Matrix Multiplication:

\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} e & f \\ g & h \end{bmatrix} = \begin{bmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{bmatrix}

Determinant:

\det(A) = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc

Statistics & Probability

Normal Distribution:

f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}

Bayes' Theorem:

P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}

Complex Analysis

Taylor Series Expansion:

The Taylor expansion expresses a holomorphic function $f(z)$ as a power series:

\displaystyle {\begin{aligned}T_{f}(z)&=\sum _{k=0}^{\infty }{\frac {(z-c)^{k}}{2\pi i}}\int _{\gamma }{\frac {f(w)}{(w-c)^{k+1}}}\,dw\\&={\frac {1}{2\pi i}}\int _{\gamma }{\frac {f(w)}{w-c}}\sum _{k=0}^{\infty }\left({\frac {z-c}{w-c}}\right)^{k}\,dw\\&={\frac {1}{2\pi i}}\int _{\gamma }{\frac {f(w)}{w-c}}\left({\frac {1}{1-{\frac {z-c}{w-c}}}}\right)\,dw\\&={\frac {1}{2\pi i}}\int _{\gamma }{\frac {f(w)}{w-z}}\,dw=f(z),\end{aligned}}

Euler's Formula:

e^{ix} = \cos(x) + i\sin(x)

Physics

Schrödinger Equation:

i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat{H}\Psi(\mathbf{r},t)

Maxwell's Equations:

\begin{aligned} \nabla \cdot \mathbf{E} &= \frac{\rho}{\epsilon_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\epsilon_0\frac{\partial \mathbf{E}}{\partial t} \end{aligned}

Lagrangian Mechanics:

The action functional $S$ is defined as:

\displaystyle S[{\boldsymbol {q}}]=\int _{a}^{b}L(t,{\boldsymbol {q}}(t),{\dot {\boldsymbol {q}}}(t))\,dt.

Advanced Features

Multi-line Equations

Use aligned environment for aligned equations:

\begin{aligned} f(x) &= (x+a)(x+b) \\ &= x^2 + (a+b)x + ab \end{aligned}

Cases and Piecewise Functions

f(x) = \begin{cases} x^2 & \text{if } x \geq 0 \\ -x^2 & \text{if } x < 0 \end{cases}

Fractions and Continued Fractions

\frac{1}{\displaystyle 1+\frac{1}{\displaystyle 2+\frac{1}{\displaystyle 3+\frac{1}{4}}}}

Greek Letters and Symbols

Common symbols used in mathematics:

Greek: $\alpha, \beta, \gamma, \delta, \epsilon, \theta, \lambda, \mu, \pi, \sigma, \omega$
Operators: $\sum, \prod, \int, \nabla, \partial$
Relations: $\leq, \geq, \neq, \approx, \equiv, \propto$
Sets: $\in, \notin, \subset, \subseteq, \cup, \cap, \emptyset$
Logic: $\forall, \exists, \neg, \land, \lor, \implies, \iff$

Subscripts and Superscripts

x_1, x_2, \ldots, x_n \quad \text{and} \quad a^2 + b^2 = c^2

Large Operators

Summation:

\sum_{i=1}^{n} i = \frac{n(n+1)}{2}

Product:

\prod_{i=1}^{n} i = n!

Integration:

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

Special Formatting

Colored Equations

KaTeX supports color through the \textcolor and \colorbox commands:

\textcolor{red}{F = ma} \quad \text{and} \quad \colorbox{yellow}{$E = mc^2$}

Sizing

Control the size of your equations:

\tiny{tiny} \quad \small{small} \quad \normalsize{normal} \quad \large{large} \quad \Large{Large} \quad \LARGE{LARGE} \quad \huge{huge}

Spacing

Fine-tune spacing in equations:

a\!b \quad a\,b \quad a\:b \quad a\;b \quad a\ b \quad a\quad b \quad a\qquad b

Best Practices

Keep It Readable

Use clear variable names and proper spacing:

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

Cramped or unclear notation:

P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}

Use Display Style for Complex Equations

For complex fractions and large operators, use \displaystyle:

\displaystyle \sum_{i=1}^{n} \frac{1}{i^2} = \frac{\pi^2}{6}

Break Long Equations

For very long equations, use multiple lines with aligned:

\begin{aligned} (a + b)^3 &= (a + b)(a + b)^2 \\ &= (a + b)(a^2 + 2ab + b^2) \\ &= a^3 + 3a^2b + 3ab^2 + b^3 \end{aligned}

Label Important Equations

Use text annotations to explain components:

\underbrace{e^{i\pi}}_{\text{Euler's identity}} + 1 = 0

Common Syntax Reference

Basic Operations

Syntax	Result	Description
`x + y`	$x + y$	Addition
`x - y`	$x - y$	Subtraction
`x \times y`	$x \times y$	Multiplication
`x \div y`	$x \div y$	Division
`\frac{x}{y}`	$\frac{x}{y}$	Fraction
`x^y`	$x^y$	Superscript
`x_y`	$x_y$	Subscript
`\sqrt{x}`	$\sqrt{x}$	Square root
`\sqrt[n]{x}`	$\sqrt[n]{x}$	nth root

Delimiters

Syntax	Result	Description
`(x)`	$(x)$	Parentheses
`[x]`	$[x]$	Brackets
`\{x\}`	$\{x\}$	Braces
`\langle x \rangle`	$\langle x \rangle$	Angle brackets
`\lvert x \rvert`	$\lvert x \rvert$	Absolute value
`\lVert x \rVert`	$\lVert x \rVert$	Norm

Troubleshooting

Equation Not Rendering

Check that katex/dist/katex.css is imported in your layout
Verify the TeX syntax is valid
Ensure remark-math and rehype-katex are configured correctly
Use the KaTeX Live Demo to test syntax

Missing Symbols

Not all LaTeX commands are supported by KaTeX
Check the KaTeX Support Table
Consider using alternative notation

Escaping Special Characters

Use backslash to escape special characters:

Use \$ for a dollar sign, not $\$$ in math mode.

Pro Tip

You can copy equations from Wikipedia - they're already in LaTeX format and work directly with KaTeX!

Try it: Visit any Wikipedia math article, right-click an equation, and select "Copy LaTeX code".

Resources

KaTeX Official Documentation
KaTeX Support Table - Complete list of supported functions
KaTeX Live Demo - Test equations in real-time
LaTeX Math Symbols - Comprehensive symbol reference
Detexify - Draw a symbol to find its LaTeX command
Fumadocs Math Guide

Next Steps

Experiment with different equation types
Check out the KaTeX support table for all available commands
Review our Mermaid Diagrams feature for visual diagrams
Explore the Development Workflow for general writing tips

Math Equations

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