Identities

Remco Bloemen

2014-05-07

The mathematical identities from my Advanced Quantum Mechanics course note:

Γ(z+1)=zΓ(z) \Gamma (z+1) = z \Gamma (z)

Γ(1ϵ)Γ(ϵ)=πsinπϵ=ϵ+π261ϵ+O(1ϵ3) \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)

0dx;xαeβx=Γ(1+α)β(1+α) \int _0^\infty \mathrm{d} x; x^\alpha \mathrm{e}^{-\beta x} = \Gamma (1+\alpha )\beta ^{-(1+\alpha )}

x1=0dα;eαx x^{-1} = \int _0^\infty \mathrm{d} \alpha ; \mathrm{e}^{-\alpha x}

0dx;xα(1x)β=Γ(1+α)Γ(1+β)/Γ(1+α+β) \int _0^\infty \mathrm{d} x; x^\alpha (1-x)^\beta = \Gamma (1+\alpha )\Gamma (1+\beta ) / \Gamma (1 + \alpha + \beta )

0da1;0dan;f(sumai)=frac1(n1)!0ds;sn1f(s) \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)