Mathematical Typesetting in LaTeX: A Comprehensive Guide

% Inline math - embedded in text
This is inline math: $E = mc^2$ and more text.

% Display math - centered, on its own line
\[
E = mc^2
\]

% Alternative display math environments
\begin{equation}
E = mc^2
\end{equation}

% Essential math packages
\usepackage{amsmath}      % Core math environments
\usepackage{amssymb}      % Additional symbols
\usepackage{amsfonts}     % Additional fonts (blackboard, script)
\usepackage{mathtools}    % Enhancements to amsmath
\usepackage{commath}      % Differential operators

% Optional but recommended
\usepackage{xfrac}        % Slanted fractions
\usepackage{physics}      # Physics notation shortcuts

% Simple fractions
\frac{a}{b}               % a/b
\frac{a + b}{c + d}       % (a+b)/(c+d)

% Nested fractions
\frac{\frac{a}{b}}{c}     % (a/b)/c
\frac{a}{\frac{b}{c}}     % a/(b/c)

% Alternative notation (xfrac package)
\sfrac{a}{b}              % Slanted fraction

% Display style fractions
\displaystyle \frac{a}{b} % Larger, display-style
\textstyle \frac{a}{b}    % Smaller, text-style

% Square roots
\sqrt{x}                  % √x
\sqrt{x^2 + y^2}          % √(x² + y²)

% nth roots
\sqrt[n]{x}               % nth root of x

% Nested roots
\sqrt{\sqrt{x}}           % √√x
\sqrt[n]{\sqrt[m]{x}}     % Nested nth and mth roots

% Exponents
x^{2}                     % x²
x^{a+b}                   % x^(a+b)
e^{-x^2}                  % e^(-x²)

% Combined
x^{2n}_{i,j}              % Subscript and superscript

% Lowercase Greek
\alpha, \beta, \gamma, \delta, \epsilon
\varepsilon, \zeta, \eta, \theta, \vartheta
\iota, \kappa, \lambda, \mu, \nu, \xi
\pi, \varpi, \rho, \varrho, \sigma, \varsigma
\tau, \upsilon, \phi, \varphi, \chi, \psi, \omega

% Uppercase Greek
\Gamma, \Delta, \Theta, \Lambda, \Xi, \Pi
\Sigma, \Upsilon, \Phi, \Psi, \Omega

% Operators
\pm, \mp                  % ±, ∓
\times, \cdot            % ×, ·
\oplus, \ominus           % ⊕, ⊖
\otimes, \oslash          % ⊗, ⊘
\sum, \prod, \int         % Σ, ∏, ∫
\partial                 % ∂
\nabla                   % ∇

% Relations
=, \neq, \approx, \equiv % =, ≠, ≈, ≡
<, >, \leq, \geq         % <, >, ≤, ≥
\subseteq, \subset      % ⊆, ⊂
\in, \notin               % ∈, ∉
\sim, \simeq             % ∼, ≃
\Rightarrow, \Leftarrow  % ⇒, ⇐
\iff                     % ⟺

% Arrows
\rightarrow, \leftarrow  % →, ←
\mapsto                   % ↦
\Longrightarrow          % ⟹
\leftrightarrow          % ↔

% Set theory
\cup, \cap                % ∪, ∩
\cupdot, \capdot          % ⊍, ⊏
\emptyset                 % ∅
\mathbb{R}, \mathbb{N}    % ℝ, ℕ (requires amsfonts)

% Logic
\forall, \exists         % ∀, ∃
\neg, \lor, \land        % ¬, ∨, ∧
\top, \bot               % ⊤, ⊥

% equation - numbered
\begin{equation}
f(x) = x^2 + 2x + 1
\end{equation}

% equation* - unnumbered (starred)
\begin{equation*}
f(x) = x^2 + 2x + 1
\end{equation*}

% \[ ... \] - unnumbered (shortcut)
\[
f(x) = x^2 + 2x + 1
\]

% gather - multiple equations, separate numbering
\begin{gather}
a = b + c \\
d = e + f
\end{gather}

% gather* - unnumbered
\begin{gather*}
a = b + c \\
d = e + f
\end{gather*}

% align - aligned equations, shared numbering
\begin{align}
a &= b + c \\
d &= e + f
\end{align}

% align - with multiple alignment points
\begin{align}
x^2 + y^2 &= 1 \\
x &= \cos\theta \\
y &= \sin\theta
\end{align}

% split - sub-numbering within equation
\begin{equation}
\begin{aligned}
a &= b + c \\
  &= d + e + f
\end{aligned}
\end{equation}

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

% With numcases (requires numcases package)
\begin{numcases}{f(x)=}
x^2 & for $x > 0$
\end{numcases}

% rcases - cases on the right side
\[
\begin{rcases}
a = b \\
c = d
\end{rcases}
\implies e = f
\]

% Basic matrix (no delimiters)
\begin{matrix}
a & b \\
c & d
\end{matrix}

% Parentheses
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}

% Brackets
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}

% Braces
\begin{Bmatrix}
a & b \\
c & d
\end{Bmatrix}

% Pipes (determinant)
\begin{vmatrix}
a & b \\
c & d
\end{vmatrix}

% Double pipes (norm)
\begin{Vmatrix}
a & b \\
c & d
\end{Vmatrix}

% Small matrix (inline)
$\begin{smallmatrix}
a & b \\ c & d
\end{smallmatrix}$

% With dots
\begin{pmatrix}
a_{11} & a_{12} & \dots \\
a_{21} & a_{22} & \dots \\
\vdots & \vdots & \ddots
\end{pmatrix}

% Row vector
\vec{v} = \begin{pmatrix} v_1 & v_2 & v_3 \end{pmatrix}

% Column vector
\vec{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}

% Arrow notation
\overrightarrow{AB}

% Bold notation
\mathbf{v}

% Hat notation (unit vectors)
\hat{v}, \hat{i}, \hat{j}, \hat{k}

\usepackage{amsthm}

% Define theorem styles
\theoremstyle{plain}       % Italic, bold header
\theoremstyle{definition} % Roman, bold header
\theoremstyle{remark}     % Roman, italic body

% Define new theorem environments
\newtheorem theorem}{Theorem}
\newtheorem*{theorem*}{Theorem}     % Unnumbered

\newtheoremlemma}{Lemma}
\newtheoremdefinition}{Definition}
\newtheoremproposition}{Proposition}
\newtheoremcorollary}{Corollary}
\newtheoremproof}{Proof}

% Numbering by section
\newtheorem{section}{Section}
\newtheorem{exercise}{Exercise}[section]

\begin{Theorem}[Pythagorean Theorem]
For a right triangle with legs $a$ and $b$ and hypotenuse $c$,
\[
a^2 + b^2 = c^2
\]
\end{Theorem}

\begin{Definition}
A \emph{prime number} is a natural number greater than 1 
that has no positive divisors other than 1 and itself.
\end{Definition}

\begin{Lemma}
If $n$ is even, then $n^2$ is even.
\end{Lemma}

\begin{Proof}
Suppose $n = 2k$ for some integer $k$. Then
\[
n^2 = (2k)^2 = 4k^2 = 2(2k^2)
\]
which is even. \qed
\end{Proof}

\theoremstyle{mystyle}
\newtheoremstyle{break}
  {\topsep}{\topsep}%
  {\itshape}{}%
  {\bfseries}{}%
  {\newline}{}%
\theoremstyle{break}
\newtheorem{breaktheorem}{Theorem}

% Basic derivatives
\frac{dy}{dx}             % dy/dx
\frac{d^2y}{dx^2}         % d²y/dx²

% Partial derivatives
\frac{\partial f}{\partial x}
\frac{\partial^2 f}{\partial x^2}

% With upright 'd' (comdiag package)
\dd{x}, \dd[2]{x}         % dx, d²x

% Using physics package
\dv{x}                    % dx
\dv[n]{x}                 % d^n x
\pdv{x}                   % ∂x
\pdv{f}{x}                % ∂f/∂x

% Leibniz notation
\left.\frac{dy}{dx}\right|_{x=0}

% Basic integrals
\int f(x) \,dx            % ∫ f(x) dx
\int_a^b f(x) \,dx        % ∫ₐᵇ f(x) dx

% Multiple integrals
\iint f(x,y) \,dx\,dy    % ∬ f(x,y) dx dy
\iiint f(x,y,z) \,dx\,dy\,dz  % ∭
\oint f(z) \,dz          % ∮

% Limits
\lim_{x \to \infty} f(x)
\liminf_{n \to \infty} a_n
\limsup_{n \to \infty} a_n

% Subscripts and superscripts
\int_{\partial M} \omega  % Integral over boundary
\int_{0}^{\infty}         % From 0 to infinity

% Summation
\sum_{i=1}^{n} a_i       % Σᵢ₌₁ⁿ aᵢ
\sum_{i \in I} a_i       % Sum over set
\sum_{n=1}^{\infty}      % Infinite sum

% Product
\prod_{i=1}^{n} a_i      % ∏ᵢ₌₁ⁿ aᵢ

% Limits
\lim_{n \to \infty} a_n  % Limit
\varliminf_{n} a_n       % Liminf
\varlimsup_{n} a_n       % Limsup

% O-notation
O(n^2)                   % Big O
o(n)                     % Small o
\Theta(n)                % Theta
\Omega(n)                % Omega
\omega(n)                % Small omega

% Vector notation
\vec{v} \in \mathbb{R}^n
\mathbf{v} \in \mathbf{R}^n

% Span
\operatorname{span}\{v_1, v_2\}
\span\{v_1, v_2\}

% Inner product
\langle u, v \rangle      % ⟨u, v⟩
(u, v)                    % Alternative
\u \cdot \v               % Dot product

% Norm
\|v\|                     % ‖v‖
\lVert v \rVert           % Alternative
\|v\|_2                   % L2 norm
\|v\|_\infty              % Infinity norm

% Matrix notation
A \in \mathbb{R}^{m \times n}
B \in \mathbb{R}^{n \times p}

% Matrix operations
A^T                      % Transpose
A^{-1}                   % Inverse
A^*                      % Adjoint/Hermitian
\det(A)                  % Determinant
\operatorname{tr}(A)     % Trace
\operatorname{rank}(A)  % Rank

% Block matrices
\begin{pmatrix}
A & B \\
C & D
\end{pmatrix}

% Identity matrix
I_n                      % n×n identity
\mathbf{I}

% Zero matrix
\mathbf{0}

% Diagonal
\operatorname{diag}(a_1, a_2, \dots, a_n)

% Pauli matrices
\sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}
\sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}
\sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}

% Divisibility
a \mid b                 % a divides b
a \nmid b                % a does not divide b

% Congruences
a \equiv b \pmod{n}     % a ≡ b (mod n)
a \equiv b \mod n       % Alternative
a \equiv b \equiv c \pmod{n}  % Chained

% GCD and LCM
\gcd(a, b)              % gcd(a,b)
\operatorname{lcm}(a,b) % lcm(a,b)

% Euler's totient
\phi(n)

% Summations over divisors
\sum_{d|n}               % Sum over divisors

% Probability
\Pr(A)                   % P(A)
\Pr(A \mid B)            % P(A|B)

% Expectations
\mathbb{E}[X]            % E[X]
\mathbb{E}[X \mid Y]     % E[X|Y]

% Variance and covariance
\operatorname{Var}(X)   % Var(X)
\operatorname{Cov}(X,Y) % Cov(X,Y)

% Distributions
N(\mu, \sigma^2)        % Normal distribution
\exp(\lambda)           % Exponential
\ Poisson(\lambda)      % Poisson
\Binomial(n, p)         % Binomial

% Probability symbols
\Omega                  % Sample space
\varnothing             % Empty set
\cup, \cap              % Union, intersection

% Using the physics package
\Bqty{a}                % Bracketed quantity
\quantity{a}           % Full quantity
\abs{x}                % |x|
\norm{x}               % ||x||
\conj{z}               % z̄ (complex conjugate)
\Re{z}                 % ℜ(z)
\Im{z}                 % ℑ(z)
\grad                  % ∇
\div                   % ∇⋅
\curl                  % ∇×

% Dirac notation
\ket{\psi}             % |ψ⟩
\bra{\psi}             % ⟨ψ|
\braket{\psi}{\phi}    % ⟨ψ|φ⟩

% Kronecker delta
\delta_{ij}

% Levi-Civita symbol
\epsilon_{ijk}

% Define custom commands
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}

% With arguments
\newcommand{\abs}[1]{\lvert#1\rvert}
\newcommand{\norm}[1]{\lVert#1\rVert}
\newcommand{\dd}[2]{\frac{d#1}{d#2}}

% Over macros
\DeclareMathOperator{\trace}{tr}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\proj}{proj}
\DeclareMathOperator{\Span}{Span}

% Thin space (normal)
a b

% Medium space
a \; b

% Thick space
a \; b

% Negative thin space
a \! b

% Quad (1em)
a \quad b

% Qquad (2em)
a \qquad b

% Using \text
f(x) = x^2 \text{ for } x > 0

% Using \mathrm
f(x) = x^2 \mathrm{~for~} x > 0

% Using \operatorname
\operatorname{arcsin}(x)

% Subequations
\begin{subequations}
\begin{align}
a &= b \\
c &= d
\end{align}
\end{subequations}

% Custom labels
\begin{equation}
E = mc^2 \tag{*} \label{eq:einstein}
\end{equation}

% References
\eqref{eq:einstein}
\tag\ref{eq:einstein}

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

\begin{equation}
\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i x \xi} \,dx
\end{equation}

\begin{equation}
f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots
     = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n
\end{equation}

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

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