How to prove beta function property: $$ \mathrm{B}(a,b) \mathrm{B}(a+b,c) = \mathrm{B}(b,c) \mathrm{B}(a,b+c) $$
using Beta function definition.
$$ \mathrm{B}(a,b) = \int_0^1x^{a-1}(1-x)^{b-1}dx. $$
2026-03-26 06:25:24.1774506324
Proving beta function property: $ B(a,b) B(a+b,c) = B(b,c) B(a,b+c) $
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The relationship between the $\Gamma$ function and the $B$ function is given by the following identity: $$ B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}. $$
Using this equality, your original equation can be re-written as $$ \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}\cdot \frac {\Gamma(a+b)\Gamma(c)}{\Gamma(a+b+c)}= \frac{\Gamma(b)\Gamma(c)}{\Gamma(b+c)}\cdot\frac{\Gamma(a)\Gamma(b+c)}{\Gamma(a+b+c)} $$ which can immediately be seen to hold.