= Complex beta integral
{title2=$\int_{\mathbb C}|z|^{2a-2}|1-z|^{2b-2}\,dx\,dy$}
For $c=1-a-b$ with positive real parts $\operatorname{Re}a,\operatorname{Re}b,\operatorname{Re}c$, the integral converges at its two finite singularities and at infinity, and equals
$$
\pi\frac{\Gamma(a)\Gamma(b)\Gamma(c)}{\Gamma(1-a)\Gamma(1-b)\Gamma(1-c)}.
$$
To prove it, use <Schwinger parameterization> with exponents $1-a$ and $1-b$. Integrating the resulting planar <Gaussian integral> gives $\pi/(\lambda+\rho)$ times $\exp[-\lambda\rho/(\lambda+\rho)]$. Set $q=\lambda+\rho$ and $x=\lambda/q$; the $q$ integral is a <gamma function> and the $x$ integral is $B(a,b)$, giving the stated ratio. This domain justifies the interchanges; elsewhere the answer is interpreted by <analytic continuation>. The string measure $d^2z=2dx\,dy$ multiplies the answer by two.
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