Identities

% Remco Bloemen % 2014-05-07

The mathematical identities from my Advanced Quantum Mechanics course note:

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

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

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

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

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

$${\int }_0^{\infty }\mathrm{d}{a}_1;\cdots {\int }_0^{\infty }\mathrm{d}{a}_n;f\left(sum{a}_i\right)=frac1(n-1)!{\int }_0^{\infty }\mathrm{d}s;s^{n-1}f(s)$$