Mathematical Expressions

% Inline math - flows with text
The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

% Display math - centered on its own line
The quadratic formula is:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

% Or using equation environment (numbered)
\begin{equation}
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\end{equation}

% Basic fraction
\frac{1}{2}

% Nested fractions
\frac{1}{1 + \frac{1}{2}}

% Display style in inline math
$\displaystyle\frac{1}{2}$

% Text style in display math
\[\textstyle\frac{1}{2}\]

% Continued fractions
\cfrac{1}{2 + \cfrac{1}{3 + \cfrac{1}{4}}}

% Binomial coefficient
\binom{n}{k} = \frac{n!}{k!(n-k)!}

% Different styles
\dbinom{n}{k}  % Display style
\tbinom{n}{k}  % Text style

% Square root
\sqrt{x}

% nth root
\sqrt[n]{x}

% Complex expressions
\sqrt{a^2 + b^2}

% Nested roots
\sqrt{\sqrt{x} + 1}

% Large expressions
\sqrt{\frac{x^2 + y^2}{2}}

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

% Display style
\displaystyle\sum_{i=1}^{n} i

% Multiple indices
\sum_{\substack{i=1 \\ j=1}}^{n} a_{ij}

% Infinite series
\sum_{n=0}^{\infty} \frac{x^n}{n!} = e^x

% Basic product
\prod_{i=1}^{n} i = n!

% Infinite product
\prod_{n=1}^{\infty} \left(1 - \frac{1}{n^2}\right)

% Co-product
\coprod_{i \in I} X_i

% Indefinite integral
\int f(x) \, dx

% Definite integral
\int_a^b f(x) \, dx

% Multiple integrals
\iint_D f(x,y) \, dx \, dy
\iiint_V f(x,y,z) \, dx \, dy \, dz

% Contour integrals
\oint_C F \cdot dr

% With limits
\int\limits_0^1 x^2 \, dx = \frac{1}{3}

% Basic limit
\lim_{x \to 0} \frac{\sin x}{x} = 1

% Limits with approach direction
\lim_{x \to 0^+} f(x)
\lim_{x \to 0^-} f(x)

% Limits to infinity
\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e

% Supremum and infimum
\sup_{x \in A} f(x)
\inf_{x \in A} f(x)

% First derivative
f'(x) \text{ or } \frac{df}{dx}

% Higher derivatives
f''(x) \text{ or } \frac{d^2f}{dx^2}

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

% With evaluation
\left.\frac{df}{dx}\right|_{x=0}

% Manual sizing
( \big( \Big( \bigg( \Bigg(

% Automatic sizing
\left( \frac{1}{2} \right)
\left[ \sum_{i=1}^n i \right]
\left\{ x : x > 0 \right\}

% Mixed delimiters
\left< \frac{a}{b} \right|
\left\lfloor x \right\rfloor
\left\lceil x \right\rceil

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

% Matrices with brackets
\begin{pmatrix}
  a & b \\
  c & d
\end{pmatrix}

% Norms and absolute values
\|x\| = \sqrt{x_1^2 + x_2^2}
|x| = \begin{cases}
  x & \text{if } x \geq 0 \\
  -x & \text{if } x < 0
\end{cases}

% Trigonometric
\sin x, \cos x, \tan x
\arcsin x, \arccos x, \arctan x

% Exponential and logarithmic
\exp(x), e^x
\log x, \ln x, \log_2 x

% Hyperbolic
\sinh x, \cosh x, \tanh x

% Other functions
\max(a,b), \min(a,b)
\gcd(a,b), \det(A)

% Define new operators
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator*{\argmax}{arg\,max}

% Usage
\sgn(x) = \begin{cases}
  1 & x > 0 \\
  0 & x = 0 \\
  -1 & x < 0
\end{cases}

\argmax_{x \in \mathbb{R}} f(x)

\begin{align}
  (a + b)^2 &= (a + b)(a + b) \\
            &= a^2 + ab + ba + b^2 \\
            &= a^2 + 2ab + b^2
\end{align}

% Without numbering
\begin{align*}
  e^{i\pi} + 1 &= 0 \\
  e^{i\pi} &= -1
\end{align*}

% Using \text
f(x) = x^2 \text{ for all } x \in \mathbb{R}

% Using \intertext in align
\begin{align}
  a &= b + c \\
  \intertext{and therefore}
  a - b &= c
\end{align}

% Math in text
The value of $\pi$ is approximately $3.14159$.