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On the complex Selberg integral. (English) Zbl 0639.33002

The complex Selberg beta integral is \[ (i/2)\quad n\int_{C\quad n}\prod_{1\leq i<j\leq n}| z_ i-z_ j|^{2\gamma}\prod^{n}_{j=1}| \quad z_ j|^{2\alpha} | z_ j-1|^{2\beta} dz_ j d\bar z_ j. \] This is evaluated as the square of the corresponding Selberg integral times appropriate trigonometric functions. The proof uses contour bending, Hadamard type finite part integrals and some homology as well as the evaluation of Selberg’s beta integral.
Reviewer: R.Askey

MSC:

33B15 Gamma, beta and polygamma functions
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