How do I demonstrate the relationship between the Beta and the Gamma function, in the cleanest way possible? I am thinking one (or two) substitution of variables is necessary, but when and how is the question.
Here is the beta function: $B(\alpha,\beta)=\int_0^1x^{\alpha-1}(1-x)^{\beta-1}dx$.
Here is the gamma function $\Gamma(\alpha)=\int_0^{\infty}t^{\alpha-1}e^{-t}dt$.
Here is the relationship between the Beta and Gamma functions: $B(\alpha,\beta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}$.
Thanks for any help!
Best Answer
Compute the double integral $$ \Gamma(\alpha)\Gamma(\beta)=\iint t^{\alpha-1}\mathrm e^{-t}s^{\beta-1}\mathrm e^{-s}\mathrm ds\mathrm dt, $$ using the change of variable $x=t/(t+s)$, $y=t+s$.