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The duplication identity isIt holds initially for $\mathrm{Re}z>0$, and then as an identity of meromorphic functions. Values at poles must be interpreted through that identity or a limit, rather than as products of finite numbers.^[ref]

A direct proof uses the equal-argument beta integral, $B(z,z)=2^{1-2z}B(z,1/2)$. Substitute the beta-gamma identity on both sides:The Gamma function has no zeros, and $\Gamma (1/2)=\sqrt{\pi }$, so cancellation yields the formula. At $z=1/2$, it gives $\Gamma (1/2)\Gamma (1)=\sqrt{\pi }$, checking the normalization. The identity is useful for converting gamma values at integer and half-integer arguments.

Write $B(n,n)={\int }_0^1[x(1-x){]}^{n-1}dx$ and set $v=2x-1$. Since $x(1-x)=(1-v^2)/4$ and $dx=dv/2$,In the last integral put $u=v^2$, giving $dv=du/(2\sqrt{u})$. This yields $B(n,n)=2^{1-2n}B(1/2,n)$, and part (a) interchanges the arguments. HenceThe equal-argument beta integral identity holds for every positive real $n$, not only positive integers.