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Let $S^{d-1}$ be the unit sphere in $\mathbb{R}^d$. Let $|x-y|$ denote the euclidean distance between to points $x$ and $y$ in $\mathbb{R}^d$.

Is there a nice expression for the following (maybe classical ?) integral, $$ I(x,y)=\int_{t\in S^{d-1}}\frac{dt}{(|t-x||t-y|)^d},\qquad|x|<1,~|y|<1. $$

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  • $\begingroup$ Do you want an analytic formula or just a upper bound? $\endgroup$
    – Liding Yao
    Commented Apr 21 at 21:20
  • $\begingroup$ I am interested in an exact formula. $\endgroup$
    – user111
    Commented Apr 21 at 21:23

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