Why is it necessary that $\operatorname{Re}(x),\operatorname{Re}(y) > 0$ for the Beta-function
$$B(x,y) = \int_0^1 t^{x-1} (1-t)^{y-1} dt$$
I suppose it is because the integral diverges when $\operatorname{Re}(x),\operatorname{Re}(y) \leq 0$? But what's so special about the threshold $\operatorname{Re}(x),\operatorname{Re}(y)=0$ ? In my eyes also $t^{-1/2}(1-t)^{-1/2}$ diverges for $t\rightarrow 0$ but fulfills the condition.
Is there an intuitive explanation or an explanation for a person not familiar with the precise notion of integrable functions or Lebesgue integration?
