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This post is a question following the previous post. In one dimensional case, we have $$ \int_0^x |y|^{1-\alpha} |x-y|^{1-\beta} d y = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} |x|^{-1+\alpha+\beta},\quad x>0 $$ for $\alpha\in [0,1[$, $\beta\in [0,1[$, and $\alpha+\beta <1$. What is interesting is that we do can take $\alpha+\beta=1$, and then the above integral is actually independent of $x$. This is the benefit of integration over $[0,x]$ instead of $R$.

My problem is then whether we could generalize the above integral to high dimensional case? In particular, by choosing $\alpha+\beta=1$, the generalized integral is expected to be independent of $x\in R^d_+$.

Thanks for any hints!


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Are you by any chance looking for this: – Suvrit Aug 10 '11 at 16:38
@Suvrit, thank you for your hint. I will check whether Selberg integral can save me instead of struggling with the beta integral. :-) – Anand Aug 10 '11 at 16:59

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