# Identities

2014-05-07

The mathematical identities from my Advanced Quantum Mechanics course note:

$\Gamma (z+1) = z \Gamma (z)$

$\Gamma (1-\epsilon ) \Gamma (\epsilon ) = \frac{\pi }{sin {\pi \epsilon }} = \epsilon + \frac{\pi ^2}{6}\frac{1}{\epsilon } + O\left(\frac{1}{\epsilon ^3}\right)$

$\int _0^\infty \mathrm{d} x; x^\alpha \mathrm{e}^{-\beta x} = \Gamma (1+\alpha )\beta ^{-(1+\alpha )}$

$x^{-1} = \int _0^\infty \mathrm{d} \alpha ; \mathrm{e}^{-\alpha x}$

$\int _0^\infty \mathrm{d} x; x^\alpha (1-x)^\beta = \Gamma (1+\alpha )\Gamma (1+\beta ) / \Gamma (1 + \alpha + \beta )$

$\int _0^\infty \mathrm{d} a_1; \cdots \int _0^\infty \mathrm{d} a_n; f\left(sum a_i \right) = frac{1}{(n-1)!} \int _0^\infty \mathrm{d} s; s^{n-1} f(s)$