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https://ello.co/elosinchryl/post/qphh__-la72opu0mustbfg
https://colab.research.google.com/drive/1iWaur0AhMxojtmYzWknQDW9UJSWQvKyk
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https://ello.co/glutexxrhin-chi/post/hqgf2judqr67s5titzes6w
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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 $\
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