What are the poles and zeros of the Euler Beta function?

Question

1. For what pairs of complex values $(x,y) \in \mathbb{C} \times \mathbb{C}$ does the Euler Beta function $B(x,y)$ equal zero?

2. For what pairs of complex values $(x,y) \in \mathbb{C} \times \mathbb{C}$ does the Euler Beta function $B(x,y)$ have a pole, and what are the orders of the poles?

Answer 1

The Beta function can be defined by $$B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$$ So it has zeroes where the denominator tends to $\infty$ in absolute value. The Gamma function tends to an infinite absolute value only when the argument is a negative integer or zero, hence we need $$x+y\in\mathbb{Z}_{\le0}$$ $$x,y\in\mathbb{C}/\mathbb{Z}_{\le0}$$ There are infinitely many such choices including $x=k\in\mathbb{C}/\mathbb{Z}_{\le0}$, $y=-k$.