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Please note: we are not saying that our file are the best downloading stuffs in the net. But we’ve set these files under the shared file system. we do not host any of the files. This file have been collected from the other servers & has been uploaded by our visitors. We do not (we never) own any of these files and we do not index any of these files. You must have explicitly & very knowingly (read carefully) download this file. In case if you noticed that any posted file has a link below and you have download that file befor. can you please inform us? Thanks for your cooperation.Q: 2-d integral of a function with a finite set of singularities Given a function $f(x,y)$ in the positive quadrant (x,y) $\in \mathbb{R}^2_+$ with regularity. For now, I assume it is smooth with respect to both x and y everywhere. Given a finite set of singularities in the xy-plane, say $S = \{(x_i, y_i)\}$. Can we prove/disprove (if not what is the counterexample) this property? Let $I = \int_{\mathbb{R}^2_+}\frac{1}{\sqrt{(x-x_i)^2+(y-y_i)^2}}dxdy$ be the 2-d Cauchy principal value integral. Finite set of singularities does NOT imply that the integral is, in general, infinity. If the function is not smooth, then the integral may be finite. In this case we may still make the bold claim that “For any sequence $x_i$ in the positive quadrant (x,y) $\in \mathbb{R}^2_+$ and any sequence $y_i$ in the positive quadrant, when $i\to\infty$, the $I_i = \int_{\mathbb{R}^2_+}\frac{1}{\sqrt{(x-x_i)^2+(y-y_i)^2}}dxdy$ tends to an integral $I$ that is independent of the sequence $x_i$ and $y_i$” A: The function $\ 1cdb36666d
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