LaTeX Mathematical Typesetting: Equations, Theorems, and Proofs

The equation $E = mc^2$ defines mass-energy equivalence.

For small values, use $\epsilon \approx 0$.

The sum $\sum_{i=1}^{n} i = n(n+1)/2$ is well-known.

% Centered, unnumbered
\[
\int_0^\infty e^{-x^2} dx = \frac{\sqrt{\pi}}{2}
\]

% Numbered equation
\begin{equation}
f(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\end{equation}

% Equation with tag
\begin{equation*}
\lim_{x \to \infty} \frac{1}{x} = 0
\end{equation*}

% Inline fraction
$\frac{a}{b}$

% Display fraction
\[\frac{a}{b}\]

% Nested fractions
\[\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{ad}{bc}\]

% Continued fraction
\[
x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3}}}
\]

$\sqrt{x}$

$\sqrt[3]{x}$

$\sqrt{x^2 + y^2}$

$\sqrt{1 + \sqrt{2}}$

$x^2$

$x^{n+1}$

$x_i$

$x_{i,j}$

${x_i}^n$ vs $x_i^n$ % Different placement

\usepackage{amsmath}

\begin{align}
x + y &= 5 \\
2x + y &= 7
\end{align}

% With numbering
\begin{align}
x + y &= 5 \\
2x + y &= 7
\end{align}

% No numbering
\begin{align*}
x + y &= 5 \\
2x + y &= 7
\end{align*}

\begin{gather}
f(x) = x^2 \\
g(x) = \sin(x) \\
h(x) = e^x
\end{gather}

\begin{equation}
\begin{split}
(a+b)^2 &= a^2 + 2ab + b^2 \\
       &= a^2 + b^2 + 2ab
\end{split}
\end{equation}

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

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

\pm, \mp, \times, \div, \cdot
\cap, \cup, \subset, \supset, \in, \notin
\wedge, \vee, \neg, \forall, \exists
\leq, \geq, \neq, \approx, \equiv, \sim
\sum, \prod, \int, \oint, \partial
\nabla, \infty, \aleph, \beth

\rightarrow, \leftarrow, \Rightarrow, \Leftarrow
\leftrightarrow, \Leftrightarrow
\mapsto, \hookleftarrow
\subset, \supset, \subseteq, \supseteq
\neq, \approx, \equiv, \cong

( ), [ ], \{ \}, | |
\left( \right)
\left[ \right]
\left\{ \right\}
\left| \right|
\left\langle \right\rangle

\[
\begin{matrix}
a & b \\
c & d
\end{matrix}
\]

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

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

\begin{vmatrix}
a & b \\
c & d
\end{vmatrix}

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

% With dots
\begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{bmatrix}

\usepackage{amsthm}

\newtheoremstyle{mystyle}
  {\topsep}   % Space above
  {\topsep}   % Space below
  {\normalfont} % Body font
  {0pt}       % Indentation
  {\bfseries} % Head font
  {.}         % Punctuation
  {5pt plus 1pt minus 1pt} % Space after head
  {}

\theoremstyle{mystyle}

\newtheoremtheorem}{theorem}{Theorem}
\newtheoremlemma}[theorem]{Lemma}
\newtheoremdefinition}{definition}{Definition}
\newtheoremproposition}{proposition}{Proposition}
\newtheoremcorollary}[theorem]{Corollary}

\begin theorem}
If $f$ is continuous on $[a,b]$, then $f$ attains its maximum value.
\end theorem}

\begin lemma}
For any $\epsilon > 0$, there exists $\delta > 0$ such that...
\end lemma}

\begin definition}
A \textbf{group} is a set $G$ with a binary operation $\cdot$ satisfying...
\end definition}

\begin proof}
We prove by induction on $n$. For $n=1$, the statement is trivial.

Assume true for $n=k$. Then for $n=k+1$...
Thus the theorem holds.
\end proof}

\numberwithin{equation}{section}
\numberwithin{equation}{subsection}

\begin{equation}
a^2 + b^2 = c^2 \tag{Pythagoras}
\end{equation}

\begin{equation}
e^{i\pi} + 1 = 0 \tag{Euler's Identity}
\end{equation}

\tag{ref:myequation}
\ref{eq:myequation}
\eqref{eq:myequation}

\!    % Negative thin space
\,    % Thin space (3/18 quad)
\:    % Medium space (4/18 quad)
\;    % Thick space (5/18 quad)
\quad % 1 quad (1 em)
\qquad % 2 quads

% Examples
\int_a^b f(x)\,dx  % Proper spacing

\, vs \,  % Nested integrals

x\,y  % Product vs xy

\text{text}

\mathrm{text}   % Roman (upright)
\mathbf{text}   % Bold
\mathsf{text}   % Sans-serif
\ittext         % Italic

\[
\text{Let } x \in \mathbb{R} \text{ be arbitrary}
\]

\left(
\begin{array}{cc}
a & b \\
c & d
\end{array}
\right)

\begin{align*}
\sqrt{1 + x} &= 1 + \frac{x}{2} - \frac{x^2}{8} + \frac{x^3}{16} - \cdots \\
             &= \sum_{n=0}^{\infty} \frac{(-1)^n (2n)!}{(1-2n)2^{2n} (n!)^2} x^n
\end{align*}

\usepackage{amscd}

\[
\begin{CD}
A @>>> B @>>> C \\
@VVV @. @VVV \\
D @>>> E @>>> F
\end{CD}
\]

% Hard to read
$$\int_0^\infty e^{-x^2}dx = \frac{\sqrt{\pi}}{2}$$

% Easier to read
\[
\int_0^\infty e^{-x^2} \, dx = \frac{\sqrt{\pi}}{2}
\]