Wikipedia:Math Sandbox

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$\int_{-\infty}^\infty e^{-x^2}\ dx = \sqrt{\pi}$

$P(r, t) = \frac{1 - r^2}{1 - 2 r \cos t + r^2}$

$\lim_{n\to\infty} {2\arctan nx\over\pi}$

$r(s) = \frac{n(s)}{n(s)+m}\langle v \rangle_{s} + \frac{m}{n(s)+m} \langle v \rangle$

$\sgn x = \lim_{n\to\infty} {2\arctan nx\over\pi}$

$\begin{matrix}ax^2 + bx + c &=& a\left(x^2 + \frac{bx}{a}\right) +c \\ &=& a\left(x^2 + \frac{bx}{a} + \left(\frac{b^2}{4a^2} - \frac{b^2}{4a^2}\right)\right) + c \\ &=& a\left(x^2+2\frac{bx}{2a}+\left(\frac{b}{2a}\right)^2\right)-\frac{b^2}{4a} +c \\ &=& a\left(x+\frac{b}{2a}\right)^2-\frac{b^2}{4a} + c \end{matrix}$