LaTeX math: Sample equations

Equations written in LaTeX, a language used specifically for creating pretty equations. Thanks be to WordPress for providing this service as part of their package!

i\hbar\frac{\partial}{\partial t}\left|\Psi(t)\right>=H\left|\Psi(t)\right>

\displaystyle\bar{M}=\frac{kT}{\Xi}\sum_N\left(\frac{\partial Q_N}{\partial D}\right)_{V,T}\lambda^N

G_{ab}^{(1)} = -\frac{1}{2}\partial^c\partial_c \bar{\gamma}_{ab} + \partial^c\partial_{(b}\bar{\gamma}_{a)c} -\frac{1}{2}\eta_{ab}\partial^c\partial^d\bar{\gamma}_{cd} = 8\pi T_{ab}

\displaystyle P_\nu^{-\mu}(z)=\frac{\left(z^2-1\right)^{\frac{\mu}{2}}}{2^\mu \sqrt{\pi}\Gamma\left(\mu+\frac{1}{2}\right)}\int_{-1}^1\frac{\left(1-t^2\right)^{\mu -\frac{1}{2}}}{\left(z+t\sqrt{z^2-1}\right)^{\mu-\nu}}dt

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