Gamma function has no zeros
= Gamma function has no zeros
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The <beta--gamma identity> gives $\Gamma(z/2)^2=B(z/2,z/2)\Gamma(z)$ for $\operatorname{Re}z>0$. A zero of $\Gamma(z)$ would therefore force zeros at every $z/2^m$. But $w\Gamma(w)=\Gamma(1+w)\to1$ as $w\to0$ in the right half-plane, a contradiction. The <Gamma function recurrence> then extends nonvanishing to every point where the meromorphic gamma function is finite. Nonpositive integers are poles, not zeros.