Timeline for Approximation of Spherical Bessel function [closed]
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| Jun 19, 2022 at 11:32 | history | edited | god_operator | CC BY-SA 4.0 |
added 78 characters in body
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| Jun 18, 2022 at 5:18 | history | closed |
ZeroTheHero Jon Custer Michael Seifert Níckolas Alves GiorgioP-DoomsdayClockIsAt-90 |
Needs details or clarity | |
| Jun 17, 2022 at 0:34 | history | edited | Níckolas Alves | CC BY-SA 4.0 |
deleted 20 characters in body; edited tags
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| Jun 13, 2022 at 11:51 | vote | accept | god_operator | ||
| Jun 12, 2022 at 17:58 | answer | added | user200143 | timeline score: 1 | |
| Jun 12, 2022 at 13:39 | comment | added | god_operator | Could you please elaborate more or provide the full working? I have tried doing the complete derivation but I do not quite get it. It would be very helpful @user200143 | |
| Jun 12, 2022 at 1:25 | comment | added | user200143 | I'm too lazy to work through it, but it appears to be a stationary phase approximation to the Bessel integral $J_m(z) = 1/(2\pi i^m) \int_0^{2\pi} d\phi e^{ix\cos\phi-im\phi}$, with $\ell j_\ell(x\ell) = \sqrt{\frac{\pi}{2x\ell}}J_{\ell+1/2}(x\ell)$. The stationary phase points are at $\sin\phi = \frac{2\ell+1}{2\ell x}$, or $\cos(\phi) = \pm\sqrt{x^2-(2\ell+1)^2/(2\ell)^2}/x \simeq \pm\frac{ \sqrt{x^2-1}}{x}$, so it looks like you might get a result proportional to $1/(x\sqrt{x^2-1})$ in the limit $x\ell$ and $\ell$ both large. | |
| Jun 11, 2022 at 11:29 | comment | added | ZeroTheHero | there is no mention of $\ell$ large in the question … | |
| Jun 11, 2022 at 9:16 | comment | added | god_operator | $l$ here is large, as I mentioned in my question, I am trying to compute de CMB power spectrum for which I consider a range of $l$ in between 2 and 1500 @ZeroTheHero | |
| Jun 10, 2022 at 15:00 | comment | added | ZeroTheHero | but what of $\ell$? It's not enough to know that $x$ is small: we also need to know if $\ell$ is also small or if it is large... what matters here is the size of $x\ell$, not the size of $x$ alone. | |
| Jun 10, 2022 at 14:39 | comment | added | god_operator | $x$ is always greater than 1 but it does not tend to infinity, it always remains small, taking a maximum value of 10 @ZeroTheHero | |
| Jun 10, 2022 at 13:25 | review | Close votes | |||
| Jun 18, 2022 at 5:18 | |||||
| Jun 10, 2022 at 13:08 | comment | added | ZeroTheHero | the question is unclear. Is this for $\ell\to\infty$ but $x\to 0$ so that $\ell x$ remains finite (for instance)? | |
| Jun 10, 2022 at 10:34 | comment | added | god_operator | Thank you very much for your comments. This mathematica notebook is from Daniel Baumann and it is the one that he suggests as simple exercise in equation 3.3.70lecture notes Baumann.@EmilioPisanty | |
| Jun 10, 2022 at 10:12 | comment | added | Emilio Pisanty | ... and, speaking of which: who is "they"? what notebook is this, and who made it? what papers does it relate to? | |
| Jun 10, 2022 at 10:12 | comment | added | Emilio Pisanty | I have never seen either form. It is definitely not applicable to the $x\to0$ limit, which means that it is presumably designed for some form of large-$x$ situation. The standard formula is this one, and it is oscillatory. If both $x$ and $l$ are large, then the oscillations could be seen as averaging out and it could be acceptable to substitute $\sin^2 \to \frac12$. But in that regime the square root thing could equally well be substituted as $\sqrt{x^2-1} \mapsto x$, so who knows what they're doing. | |
| S Jun 10, 2022 at 9:43 | history | edited | god_operator | CC BY-SA 4.0 |
improved grammar
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| S Jun 10, 2022 at 9:43 | history | suggested | Brendan Darrer | CC BY-SA 4.0 |
improved grammar
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| Jun 10, 2022 at 9:41 | review | Suggested edits | |||
| S Jun 10, 2022 at 9:43 | |||||
| Jun 10, 2022 at 9:18 | history | asked | god_operator | CC BY-SA 4.0 |