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

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?

• What did you try? – Ultradark May 7 at 20:26
• My first guess would be to use the expression in term of $\Gamma$ and things like the reflection formula ($\Gamma(s)\Gamma(1-s)-\frac{\pi}{\sin \pi s}$ is entire and bounded) to find its zeros/poles, if someone knows a more elementary way I would be glad to know it. Also the zeros/poles of a meromorphic function of two complex variables are (almost) analytic curves, not isolated points. – reuns May 7 at 20:34

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 $$x=k\in\mathbb{C}/\mathbb{Z}_{\le0}$$, $$y=-k$$.