Identities

% Remco Bloemen % 2014-05-07

The mathematical identities from my Advanced Quantum Mechanics course note:

Γ(z+1) = z Γ(z)

Γ(1-ε) Γ(ε) = \frac{π}{sin {πε}} = ε + \frac{π^2}{6}\frac{1}{ε} + O\left(\frac{1}{ε^3}\right)

∫_0^∞ \mathrm{d} x; x^α \mathrm{e}^{-βx} = Γ(1+α)β^{-(1+α)}

x^{-1} = ∫_0^∞ \mathrm{d} α; \mathrm{e}^{-αx}

∫_0^∞ \mathrm{d} x; x^α (1-x)^β = Γ(1+α)Γ(1+β) / Γ(1 + α + β)

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