Timeline for Approximation for a Bessel function integral
Current License: CC BY-SA 4.0
15 events
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| Jun 25, 2023 at 12:20 | comment | added | Iosif Pinelis | @mzw : At least, can we assume that $\Delta/\rho_0<<1$? (Which would be a meaningful replacement of the meaningless condition $\Delta<<1$.) | |
| Jun 25, 2023 at 9:43 | history | edited | YCor |
edited tags
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| Jun 25, 2023 at 2:24 | comment | added | Iosif Pinelis | @mzw : As noted before, your expression for $p_{tot}$ is incorrect. It should be clear that $p_{tot}$ is a function of the dimensionless quantities $\rho_0/\Delta$ and $r/\Delta$, where $r:=R/D$. So, the conditions $R/D<<1$ and $\Delta<<1$ make no sense, because $R/D$ and $\Delta$ are not dimensionless and therefore cannot be compared to $1$. To get a definite asymptotic, you need to say about each of the three ratios -- $\rho_0/\Delta$, $r/\Delta$, and $\rho_0/r$ (or at least two relevant ratios -- say $\rho_0/\Delta$ and $r\rho_0/\Delta^2$) whether it is $>>1$, $<<1$, or $\asymp1$. | |
| Jun 24, 2023 at 10:25 | comment | added | Carlo Beenakker | @mzw : I added the plots for the small-$\rho_0$ approximation; as you can see, you don't really need $\rho_0$ to be "very" small or large. | |
| Jun 24, 2023 at 5:52 | comment | added | mzw | $\rho_0$ can be very small or very large. Anyway I think from the discussions here I figured out that I can probably get a good solution by Taylor expansion of I0 for small $\rho_0$, and by doing the replacement with $e^x/\sqrt{2 \pi x}$ and Taylor expansion of $\sqrt{\rho}$ for large $\rho_0$. So, thanks for your help. | |
| Jun 23, 2023 at 20:34 | comment | added | Iosif Pinelis | Without (I) and (II), there are a number of different kinds of asymptotics. I am not sure there will be someone willing to spend time to adequately cover all those cases. Concerning (II), can we at least assume that $\rho_0$ is on the order of $1$ (not $<<1$ and not $>>1$)? | |
| Jun 23, 2023 at 20:24 | comment | added | mzw | (i) No. (ii) No, I need a range of values for $\rho_0$ (iii) There may be a factor missing that's not really relevant for the Bessel integral. (iv) yes. | |
| Jun 23, 2023 at 19:56 | comment | added | Iosif Pinelis | (I) Can it be assumed that $R/D$ is much greater than $\Delta^2$ (and $R/D$ is much less than $1$)? (II) Can $\rho_0>0$ be assumed fixed? (III) Your expression for $p_{tot}$ is incorrect. (IV) Does your symbol $<<$ mean "much less"? | |
| Jun 23, 2023 at 18:26 | comment | added | mzw | (i) The integral is over $\int \int p(\theta,\phi) d\theta d\phi$ and the limits are a circle with radius $R/D$ around (0,0). (ii) D is in the upper limit of the integral. (iii) R/D <<1 (iv) sorry, physicist (v) and (vi) $\Delta << 1$ | |
| Jun 23, 2023 at 12:12 | comment | added | Iosif Pinelis | (v) How can angles, with values (say) in $[0,2\pi)$, be normally distributed? (vi) Is $\Delta$ small? | |
| Jun 23, 2023 at 11:44 | comment | added | Iosif Pinelis | (i) What is actually your integral in terms of $\phi$ and $\theta$? (ii) Why does $D$ not enter your expression for $p_{tot}$? (iii) What do you mean by "far away distance $D$? That $D\to\infty$ (iv) Can you state your question formally, without darts, " far away distance", ...? | |
| Jun 23, 2023 at 10:25 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
more informative title
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| Jun 23, 2023 at 9:03 | answer | added | Carlo Beenakker | timeline score: 1 | |
| S Jun 23, 2023 at 5:39 | review | First questions | |||
| Jun 23, 2023 at 6:20 | |||||
| S Jun 23, 2023 at 5:39 | history | asked | mzw | CC BY-SA 4.0 |