Timeline for A bound for the Bessel function of the first kind J_0
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 17, 2023 at 18:53 | comment | added | van der Wolf | what do you mean @username? | |
| Oct 16, 2023 at 13:04 | comment | added | username | Since @hoboonsuan points out that a sharper bound is available, for example (from his remarks) the function $$x\to \min\left(\sqrt{\frac{2}{\pi x}},\frac13 + \frac23 \cos\left(\min\left(x,\frac{\pi}{2}\right)\sqrt{\frac34}\right)\right)$$ why would you want to publish a worse bound? | |
| Jun 14, 2023 at 10:21 | comment | added | van der Wolf | Thank you, I do have a proof of the statement (quite short, using Krasikov, I. (2006). Uniform Bounds for Bessel Functions), but I'm wondering if it was proven already an published somewhere in this or equivalent form... | |
| Jun 13, 2023 at 13:26 | comment | added | ho boon suan | To handle $x$ close to the origin, we can then use the bound $J_0(x)\le{1\over3}(1+2\cos(x\sqrt{3/4}))$ for $|x|\le\pi/2$, which follows from Theorem 2.2 in Edward Neuman, “Inequalities Involving Bessel Functions of The First Kind,” Journal of Inequalities in Pure and Applied Mathematics 5 (4) (2004), Article 94; see emis.de/journals/JIPAM/article449.html | |
| Jun 13, 2023 at 12:47 | comment | added | ho boon suan | You may already know this, but I note that the bound $|J_0(x)|\le\min(1,\sqrt{2/(\pi x)})$ is known; it is weaker than your bound near the origin, for $|x|<2/\sqrt{\pi^2-4}$. See math.stackexchange.com/q/1447137 | |
| Jun 13, 2023 at 9:29 | history | asked | van der Wolf | CC BY-SA 4.0 |