Whenever you typeset mathematical notation, it needs to have “Math” style. For example: If \(a\) is an integer, then \(2a+1\) is odd.
Superscripts and subscripts are created using the characters ^ and _, respectively: \(x^2+y^2=1\) and \(a_n=0\). It is fine to have both on a single letter: \(x_0^2\).
If the superscript [or subscript] is more than a single character, enclose the superscript in curly braces: \(e^{-x}\).
Greek letters are typed using commands such as \gamma (\(\gamma)\) and \Gamma (\(\Gamma\)).
Named mathematics operators are usually typeset in roman. Most of the standards are already available. Some examples: \(\det A\), \(\cos\pi\), and \(\log(1-x)\).
When an equation becomes too large to run in-line, you display it in a “Math” paragraph by itself.
\[ f(x) = 5x^{10}-9x^9 + 77x^8 + 12x^7 + 4x^6 - 8x^5 + 7x^4 + x^3 -2x^2 + 3x + 11. \]
The \begin{aligned}...\end{aligned} environment is superb for lining up equations.
\[ \begin{aligned} (x-y)^2 &= (x-y)(x-y) \\ &= x^2 -yx - xy + y^2 \\ &= x^2 -2xy +y^2. \end{aligned} \]
\[ \begin{aligned} 3x-y&=0 & 2a+b &= 4 \\ x+y &=1 & a-3b &=10 \end{aligned} \]
To insert ordinary text inside of mathematics mode, use \text:
\[ f(x) = \frac{x}{x-1} \text{ for $x\not=1$}. \]
This is the \(3^{\text{rd}}\) time I’ve asked for my money back.
The \begin{cases}...\end{cases} environment is perfect for defining functions piecewise:
\[ |x| = \begin{cases} x & \text{when $x \ge 0$ and} \\ -x & \text{otherwise.} \end{cases} \]
Equality-like: \(x=2\), \(x \not= 3\), \(x \cong y\), \(x \propto y\), \(y\sim z\), \(N \approx M\), \(y \asymp z\), \(P \equiv Q\).
Order: \(x < y\), \(y \le z\), \(z \ge 0\), \(x \preceq y\), \(y\succ z\), \(A \subseteq B\), \(B \supset Z\).
Arrows: \(x \to y\), \(y\gets x\), \(A \Rightarrow B\), \(A \iff B\), \(x \mapsto f(x)\), \(A \Longleftarrow B\).
Set stuff: \(x \in A\), \(b \notin C\), \(A \ni x\). Use \notin rather than \not\in. \(A \cup B\), \(X \cap Y\), \(A \setminus B = \emptyset\).
Arithmetic: \(3+4\), \(5-6\), \(7\cdot 8 = 7\times8\), \(3\div6=\frac{1}{2}\), \(f\circ g\), \(A \oplus B\), \(v \otimes w\).
Mod: As a binary operation, use \bmod: \(x \bmod N\). As a relation use \mod, \pmod, or \pod:
\[ \begin{aligned} x &\cong y \mod 10 \\ x &\cong y \pmod{10} \\ x &\cong y \pod{10} \end{aligned} \]
Calculus: \(\partial F/\partial x\), \(\nabla g\).
Do not type three periods; instead use \cdots between operations and \ldots in lists: \(x_1 + x_2 + \cdots + x_n\) and \((x_1,x_2,\ldots,x_n)\).
Fractions: \(\frac{1}{2}\), \(\frac{x-1}{x-2}\).
Binomial coefficients: \(\binom{n}{2}\).
Sums and products. Do not use \Sigma and \Pi.
\[ \sum_{k=0}^\infty \frac{x^k}{k!} \not= \prod_{j=1}^{10} \frac{j}{j+1}. \]
\[ \bigcup_{k=0}^\infty A_k \qquad \bigoplus_{j=1}^\infty V_j \]
Integrals:
\[ \int_0^1 x^2 \, dx \]
The extra bit of space before the \(dx\) term is created with the \, command.
Limits:
\[ \lim_{h\to0} \frac{\sin(x+h) - \sin(x)}{h} = \cos x . \]
Also \(\limsup_{n\to\infty} a_n\).
Radicals: \(\sqrt{3}\), \(\sqrt[3]{12}\), \(\sqrt{1+\sqrt{2}}\).
Matrices:
\[ A = \left[\begin{matrix} 3 & 4 & 0 \\ 2 & -1 & \pi \end{matrix}\right] . \]
A big matrix:
\[ D = \left[ \begin{matrix} \lambda_1 & 0 & 0 & \cdots & 0 \\ 0 & \lambda_2 & 0 & \cdots & 0 \\ 0 & 0 & \lambda_3 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & \lambda_n \end{matrix} \right]. \]
Parentheses and square brackets are easy: \((x-y)(x+y)\), \([3-x]\).
For curly braces use \{ and \}: \(\{x : 3x-1 \in A\}\).
Absolute value: \(|x-y|\), \(|\vec{x} - \vec{y}|\).
Floor and ceiling: \(\lfloor \pi \rfloor = \lceil e \rceil\).
To make delimiters grow so they are properly sized to contain their arguments, use \left and \right:
\[ \left[ \sum_{n=0}^\infty a_n x^n \right]^2 = \exp \left\{ - \frac{x^2}{2} \right\} \]
Occasionally, it is useful to coerce a larger sized delimiters than \left/\right produce. Look at the two sides of this equation:
\[ \left((x_1+1)(x_2-1)\right) = \bigl((x_1+1)(x_2-1)\bigl). \]
I think the right is better. Use \bigl, \Bigl, \biggl, and the matching \bigr, etc.
Underbraces:
\[ \underbrace{1+1+\cdots+1}_{\text{$n$ times}} = n . \]
Primes: \(a'\), \(b''\).
Hats: \(\bar a\), \(\hat a\), \(\vec a\), \(\widehat{a_j}\).
Vectors are often set in bold: \(\mathbf{x}\).
Calligraphic letters (for sets of sets): \(\mathcal{A}\).
Blackboard bold for number systems: \(\mathbb{C}\).
The text above is based on a paper by Edward R. Scheinerman1.
A few more examples from mathTeX tutorial2.
\[ e^x=\sum_{n=0}^\infty\frac{x^n}{n!} \]
\[ e^x=\lim_{n\to\infty} \left(1+\frac xn\right)^n \]
\[ \varepsilon = \sum_{i=1}^{n-1} \frac1{\Delta x} \int\limits_{x_i}^{x_{i+1}} \left\{ \frac1{\Delta x}\big[ (x_{i+1}-x)y_i^\ast+(x-x_i)y_{i+1}^\ast \big]-f(x) \right\}^2dx \]
Solution for quadratic:
\[ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \]
Definition of derivative:
\[ f^\prime(x)\ = \lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x} \]
Continued fraction:
\[ f=b_o+\frac{a_1}{b_1+\frac{a_2}{b_2+\frac{a_3}{b_3+a_4}}} \]
Demonstrating \left\{…\right. and accents.
\[ \tilde y=\left\{ {\ddot x \mbox{ if $x$ odd}\atop\widehat{\bar x+1}\text{ if even}}\right. \]
Overbrace and underbrace:
\[ \overbrace{a,...,a}^{\text{k a's}}, \underbrace{b,...,b}_{\text{l b's}}\hspace{10pt} \underbrace{\overbrace{a...a}^{\text{k a's}}, \overbrace{b...b}^{\text{l b's}}}_{\text{k+l elements}} \]
Illustrating array:
\[ A\ =\ \left( \begin{array}{c|ccc} & 1 & 2 & 3 \\ \hline 1&a_{11}&a_{12}&a_{13} \\ 2&a_{21}&a_{22}&a_{23} \\ 3&a_{31}&a_{32}&a_{33} \end{array} \right) \]
See Wikibook on LaTeX for more examples.