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  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?

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  • $\begingroup$ What did you try? $\endgroup$ – Ultradark May 7 at 20:26
  • $\begingroup$ 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. $\endgroup$ – reuns May 7 at 20:34
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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 $x=k\in\mathbb{C}/\mathbb{Z}_{\le0}$, $y=-k$.

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