Sum-and-ratio substitution for gamma integrals
= Sum-and-ratio substitution for gamma integrals
For $s,t>0$, put $r=s+t$ and $u=t/(s+t)$. Then $(s,t)=(r(1-u),ru)$ and $ds\,dt=r\,dr\,du$, separating radial gamma and ratio beta integrals.
= Sum-and-ratio substitution for gamma integrals
For $s,t>0$, put $r=s+t$ and $u=t/(s+t)$. Then $(s,t)=(r(1-u),ru)$ and $ds\,dt=r\,dr\,du$, separating radial gamma and ratio beta integrals.