Timeline for Verify $ \limsup_{\epsilon \rightarrow 0^+} \int_{D}\frac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}\frac{1}{\sqrt{1-\sqrt{x^2+y^2}}} \, dx \, dy <+\infty$
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 21 at 8:33 | vote | accept | Jessi | ||
| Aug 18 at 3:45 | answer | added | Michael Engelhardt | timeline score: 6 | |
| Aug 16 at 23:26 | comment | added | Christian Remling | I think integrability at $\epsilon =0$ (not exactly what the OP is asking, of course) can be established by changing variables to $(x,y)=(s+1,y)$ and doing a Taylor expansion of the denominator of the second factor about $(s,y)=(0,0)$ (obviously the only point that poses problems). When we then integrate in polar coordinates (wrt $(s,y)$), we should be dealing with essentially $\int d\varphi\int dr\,/\sqrt{r\cos\varphi +(r^2/2)\sin\varphi}$, and now the $r$ integral is of order $\log \cos\varphi$ near $s=0$, which is integrable in $\varphi$. | |
| Aug 16 at 0:50 | comment | added | David E Speyer | I'm sorry, I screwed up and dropped a 2, and my answer is unfixable. I still think the integral is convergent, and I might try and leave another answer, but I'm going to leave this alone for a few days and hope someone else will answer it in the meantime. | |
| Aug 15 at 23:55 | comment | added | David E Speyer | Post deleted until I figure out whether Patrick Li's objection is fatal. | |
| Aug 15 at 22:24 | history | edited | Michael Hardy | CC BY-SA 4.0 |
added 7 characters in body; edited title
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| Aug 15 at 10:46 | review | Close votes | |||
| Aug 15 at 18:46 | |||||
| Aug 15 at 10:37 | history | edited | LSpice | CC BY-SA 4.0 |
No `\dfrac` in title; deleted "Thank you"
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| S Aug 15 at 10:11 | review | First questions | |||
| Aug 15 at 12:48 | |||||
| S Aug 15 at 10:11 | history | asked | Jessi | CC BY-SA 4.0 |