Mathematical Equations

\documentclass{article}
\usepackage{amsmath}
\begin{document}

% Simple display equation
\[ E = mc^2 \]

% With equation number
\begin{equation}
  E = mc^2
\end{equation}

% Without equation number
\begin{equation*}
  E = mc^2
\end{equation*}

% Reference an equation
\begin{equation}\label{eq:energy}
  E = mc^2
\end{equation}
As shown in Equation \ref{eq:energy}, energy and mass are related.

% Using \eqref for parentheses
As shown in Equation \eqref{eq:energy}...

\end{document}

% Suppress numbering for specific equation
\begin{equation}
  a^2 + b^2 = c^2 \nonumber
\end{equation}

% Custom numbering
\begin{equation}
  F = ma \tag{Newton's 2nd Law}
\end{equation}

% Numbered with custom tag
\begin{equation}
  e^{i\pi} + 1 = 0 \tag{$\star$}
\end{equation}

% Subequations
\begin{subequations}
\begin{equation}
  x + y = 5
\end{equation}
\begin{equation}
  2x - y = 1
\end{equation}
\end{subequations}

\documentclass{article}
\usepackage{amsmath}
\begin{document}

\begin{equation}
\begin{split}
  (a + b)^4 &= (a + b)^2(a + b)^2 \\
            &= (a^2 + 2ab + b^2)(a^2 + 2ab + b^2) \\
            &= a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4
\end{split}
\end{equation}

% Left-aligned split
\begin{equation}
\begin{split}
  \text{LHS} &= \text{some long expression} \\
  &\quad + \text{continuation} \\
  &\quad + \text{more terms} \\
  &= \text{RHS}
\end{split}
\end{equation}

\end{document}

\begin{multline}
  \int_0^1 \biggl\{ \sum_{i=1}^n x_i^2 + \sum_{j=1}^m y_j^2 
  + \sum_{k=1}^p z_k^2 \biggr\} \, dx \\
  + \int_1^2 \biggl\{ \sum_{i=1}^n x_i^3 + \sum_{j=1}^m y_j^3 
  + \sum_{k=1}^p z_k^3 \biggr\} \, dx \\
  = \text{some complicated result}
\end{multline}

% Control positioning
\begin{multline}
  \text{First line flush left} \\
  \text{Middle lines centered} \\
  \shoveleft{\text{This line shoved left}} \\
  \shoveright{\text{This line shoved right}} \\
  \text{Last line flush right}
\end{multline}

\documentclass{article}
\usepackage{amsmath}
\begin{document}

% Basic alignment
\begin{align}
  2x + 3y &= 7 \\
  5x - 2y &= 4
\end{align}

% Multiple alignment points
\begin{align}
  x &= a + b     &\qquad y &= c + d \\
  2x &= 2(a + b) &\qquad 3y &= 3(c + d)
\end{align}

% Without numbering
\begin{align*}
  \sin^2\theta + \cos^2\theta &= 1 \\
  1 + \tan^2\theta &= \sec^2\theta \\
  1 + \cot^2\theta &= \csc^2\theta
\end{align*}

% Selective numbering
\begin{align}
  a &= b + c \\
  d &= e + f \nonumber \\
  g &= h + i
\end{align}

\end{document}

% For alignment within a single equation number
\begin{equation}
\begin{aligned}
  f(x) &= (x+a)(x+b) \\
       &= x^2 + (a+b)x + ab
\end{aligned}
\end{equation}

% Multiple aligned blocks
\begin{equation}
\left\{
\begin{aligned}
  x + y + z &= 1 \\
  2x - y + 3z &= 0 \\
  x - 2y - z &= 4
\end{aligned}
\right.
\end{equation}

\begin{equation}
\begin{array}{lcl}
  f(x) &=& (x+1)^2 \\
       &=& x^2 + 2x + 1
\end{array}
\end{equation}

% Multiple columns
\begin{equation}
\begin{array}{ccc}
  a_{11} & a_{12} & a_{13} \\
  a_{21} & a_{22} & a_{23} \\
  a_{31} & a_{32} & a_{33}
\end{array}
\end{equation}

\documentclass{article}
\usepackage{amsmath}
\begin{document}

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

% Multiple conditions
\begin{equation}
\text{sgn}(x) = \begin{cases}
  1 & \text{if } x > 0 \\
  0 & \text{if } x = 0 \\
  -1 & \text{if } x < 0
\end{cases}
\end{equation}

% Nested cases
\begin{equation}
f(x,y) = \begin{cases}
  \begin{cases}
    1 & \text{if } y > x \\
    0 & \text{if } y = x
  \end{cases} & \text{if } x \geq 0 \\
  -1 & \text{if } x < 0
\end{cases}
\end{equation}

% Left cases
\begin{equation}
\begin{rcases}
  x^2 + y^2 = 1 \\
  x + y = 0
\end{rcases} \text{defines a curve}
\end{equation}

\end{document}

% Multiple centered lines with one number
\begin{equation}
\begin{gathered}
  (a + b)^2 = a^2 + 2ab + b^2 \\
  (a - b)^2 = a^2 - 2ab + b^2 \\
  (a + b)(a - b) = a^2 - b^2
\end{gathered}
\end{equation}

% Within text
The identities $\begin{gathered}
  \sin^2\theta + \cos^2\theta = 1 \\
  \tan\theta = \frac{\sin\theta}{\cos\theta}
\end{gathered}$ are fundamental.

% Aligning equals signs and operators
\begin{align}
  f(x) &= x^2 + 2x + 1 \\
       &= (x + 1)^2 \\
       &> 0 \quad \text{for all } x \neq -1
\end{align}

% Multiple columns
\begin{align}
  a_1 &= b_1 + c_1 & a_2 &= b_2 + c_2 \\
  d_1 &= e_1 + f_1 & d_2 &= e_2 + f_2
\end{align}

% Alignment with text
\begin{align}
  2x + 3 &= 7 & &\text{(given)} \\
  2x &= 4 & &\text{(subtract 3)} \\
  x &= 2 & &\text{(divide by 2)}
\end{align}

\begin{align}
  a &= b + c \\
  &= d + e + f
\intertext{Now we substitute the known values:}
  &= 1 + 2 + 3 \\
  &= 6
\end{align}

% Shorter spacing
\begin{align}
  x^2 + y^2 &= r^2
\shortintertext{and}
  x &= r\cos\theta \\
  y &= r\sin\theta
\end{align}

% Simple box
\begin{equation}
\boxed{E = mc^2}
\end{equation}

% Colored box
\usepackage{xcolor}
\begin{equation}
\colorbox{yellow}{$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$}
\end{equation}

% Framed important result
\begin{equation}
\boxed{
  \int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}
}
\end{equation}

% Highlight part of equation
\begin{equation}
f(x) = \underbrace{x^2 + 2x}_{\text{quadratic}} + 
       \overbrace{3x + 4}^{\text{linear}}
\end{equation}

% Spacing commands
\begin{align}
  f(x) &= x^2+2x+1 \\              % No space
  f(x) &= x^2 + 2x + 1 \\          % Normal space
  f(x) &= x^2\,+\,2x\,+\,1 \\      % Thin space
  f(x) &= x^2\:+\:2x\:+\:1 \\      % Medium space
  f(x) &= x^2\;+\;2x\;+\;1 \\      % Thick space
  f(x) &= x^2\quad+\quad2x\quad+\quad1  % Quad space
\end{align}

% Negative space
\begin{equation}
  \int\!\!\!\int f(x,y) \, dx \, dy  % Bring integral signs closer
\end{equation}

% Using array
\begin{equation}
\left\{
\begin{array}{rcrcrcr}
  2x & + & 3y & - &  z & = & 4 \\
   x & - &  y & + & 2z & = & 1 \\
  3x & + &  y & - &  z & = & 0
\end{array}
\right.
\end{equation}

% Using aligned
\begin{equation}
\left\{
\begin{aligned}
  2x + 3y - z &= 4 \\
  x - y + 2z &= 1 \\
  3x + y - z &= 0
\end{aligned}
\right.
\end{equation}

\begin{align}
  (x + y)^2 &= (x + y)(x + y) \\
            &= x(x + y) + y(x + y) \\
            &= x^2 + xy + yx + y^2 \\
            &= x^2 + 2xy + y^2
\end{align}