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\[xy^2\]
\[\sqrt[3]{x}\]
\[x_{23x^2y}y^3 \sin x \frac{x^2}{yx_3} \sqrt[3]{x}\]
Some text with \[x^2_2\] equations
\[\sin{}x\]
e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = \lim_{n\rightarrow\infty} (1+x/n)^n
\int_{0}^{1} x dx = \left[ \frac{1}{2}x^2 \right]_{0}^{1} = \frac{1}{2}
L = \int_a^b \left( g_{\it ij} \dot u^i \dot u^j \right)^{1/2} dt
\iiint f(x,y,z)\,dx\,dy\,dz
\lim_{\substack{x\rightarrow 0\\y\rightarrow 0}} f(x,y)
\idotsint_{\textrm{paths}} \exp{(iS(x,\dot{x})/\hbar)}\, \mathcal{D}x
A \alpha B \beta \Gamma \gamma \Delta \delta \dots \Phi \phi X \chi \Psi \psi \Omega \omega
\Gamma^l_{ki} = \frac{1}{2} g^{lj} (\partial_k g_{ij} + \partial_i g_{jk} - \partial_j g_{ki})
\sigma_{3} = \left( \begin{array*}{cc} 1 & 0\\ 0 & -1 \end{array*} \right)
\begin{align*} u &= \ln x \quad & dv &= x\,dx \\ du &= \mbox{$\frac{1}{x}\,dx$} & v &= \mbox{$\frac{1}{2} x^2$} \end{align*}
\newcommand{\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } } i \hbar \pd{\Psi}{t}{} = - \frac{\hbar^2}{2 m} \ \pd{\Psi}{x}{2} + V \Psi
\newcommand{\mean}[1]{{<\!\!{#1}\!\!>}} \newcommand{\braket}[2]{{<\!\!{#1|#2}\!\!>}} \newcommand{\braketop}[3]{{<\!\!{#1|\hat{#2}|#3}\!\!>}} \braket{\phi}{\psi} \equiv \int \phi^*(x) \psi(x)\,dx
\begin{array*}{l | c|c|c|c |} & \overline{A}\,\overline{B}&A\,\overline{B}&\overline{A}\, B&A\, B\\ \hline \overline{C}&0&1&0&0\\ \hline C&1&0&1&1\\ \hline \end{array*}
\begin{equation*} \begin{split} \tau &= \tau_1+\tau_2 = \sqrt{{\Delta t_1}^2-{\Delta x_1}^2}+ \sqrt{{\Delta t_2}^2-{\Delta x_2}^2} \\ &= \sqrt{(5-0)^2-(4-0)^2}+\sqrt{(10-5)^2-(0-4)^2}\\ &= 3+3 = 6 \end{split} \end{equation*}
\Gamma^l_{ki} = \frac{1}{2} g^{lj} (\partial_k g_{ij} + \partial_i g_{jk} - \partial_j g_{ki})
{\cal L}_R = \sum_{i=1}^G \bar{E}^i_R(i / \partial - g_1Y_E / B)E^i_R + \bar{D}^i_R(i / D - g_1Y_D / B)D^i_R + \bar{U}^i_R(i / D - g_1Y_U / B)U^i_R
V = \left( \begin{array}{ccc} 1-\frac{1}{2}\lambda^2 & \lambda & A\lambda^3(\rho-i\eta) \\ -\lambda & 1-\frac{1}{2}\lambda^2 & A\lambda^2 \\ A\lambda^3(1-\rho-i\eta) & -A\lambda^2 & 1 \end{array} \right) + {\cal O}(\lambda^4)
H_1 = - {\sin \phi - \sin \phi_{\rm s}\over \beta c} \, \sum_{j} e V_j \delta (\theta -\theta_j) \left(D\,{p_x\over p_0} - D'x \right)
i\hbar\frac{\partial\Psi}{\partial t} = \frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} + V\Psi