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I am interested in showing that the integral \begin{align} & \int_{\mathbb{C}} |z|^{2\alpha-2}|1-z|^{2\beta - 2} \,dA(z) \\[8pt] = {} & \int_{\mathbb{R}^2} (x^2+y^2)^{\alpha-1}((1-x)^2+y^2)^{\beta-1} \,dx\,dy \\[8pt] = {} & \frac{\sin(\pi \alpha)\sin(\pi \beta)}{\sin(\pi(\alpha+\beta))}B(\alpha,\beta)^2 \end{align} for $\Re{(\alpha)}, \Re{(\beta)} >0, \Re(\alpha+\beta)<1$, where $B$ is the usual beta function.

It is a special case of the complex Selberg integral calculated by K. Aomoto. In his paper, he makes use of homology theory but states that this special case was calculated by R. S. Strichartz using a strictly analytical argument. The reference for this was a preprint 'Complex hypergeometric integrals' 1985 which I am unable to find. I am looking for either a proof using minimal homology theory or a reference.

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    $\begingroup$ Your integral can be written as $$\int_0^{2\pi} \int_0^\infty \frac{r^{2\alpha-1}}{\left(1-2r\cos(\theta)+r^2\right)^{1-\beta}}\,dr\,d\theta.$$ $\endgroup$
    – Dispersion
    Commented Aug 7, 2023 at 15:56
  • $\begingroup$ probably related mathoverflow.net/questions/394010/… $\endgroup$
    – Abdelmalek Abdesselam
    Commented Aug 8, 2023 at 19:29

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