The Euler Beta function is defined by $$B(x,y):=\int_0^1t^{x-1}(1-t)^{y-1}dt.$$ To show that B is well defined, i do the follwing steps, so please tell me if they are true or not !
- if $0\leq t\leq 1/2$, then $(1-t)^{y-1} \leq 2^{1-y}$, for all $y\in \mathbb{R}$, or $\int_0^{1/2}t^{x-1}dt$ converges if $x>0$, therefore $\int_0^{1/2}t^{x-1}(1-t)^{y-1}dt$ converges if $x>0$.
- if $1/2 \leq t \leq 1$, then $t^{x-1} \leq 2^{1-x}$ and $\int_{1/2}^1(1-t)^{y-1}dt=\int_0^{1/2}u^{y-1}du \quad (u=1-t)$ converges if $y>0$, which implies $\int_{1/2}^1t^{x-1}(1-t)^{y-1}dt$ converges if $y>0$.
Conclusion : By using chasles relation, we have for all $x>0$ and $y>0$, $\int_0^1 t^{x-1}(1-t)^{y-1}dt$ converges.