Timeline for Is it possible to sum this analytically in any way?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 5, 2022 at 13:49 | comment | added | CfourPiO | Thank you! I appreciate it. | |
| Dec 5, 2022 at 13:43 | comment | added | Carlo Beenakker | the integrand can be worked out as a product of $e^{-x^2/2}$ times a sum of exponentials $e^{ipx/2}$, with integer $p$, which can then be integrated term by term. | |
| Dec 5, 2022 at 13:40 | comment | added | CfourPiO | I was referring to the integral in your answer. Yes, the expression is very nice and can be evaluated for any $N$. I was just curious how this expression is derived from the integral. Is it manually done by parts? | |
| Dec 5, 2022 at 13:36 | comment | added | Carlo Beenakker | which integral are you referring to? the integral in my answer can be readily evaluated for any given $N$; presumably if you want to ask a new question you will want to start a new post (one question per post is the general principle here) | |
| Dec 5, 2022 at 13:29 | comment | added | CfourPiO | I again looked at my physical problem and found that I do not need the cosine term in the expression at all. So, the expectation of $L$ is a summation over the Dirictlet Kernel function ($\sin(N x/2)/\sin(x/2)$) when $x$ is Gaussian distributed. I actually tried solving the integral that you suggested in the answer on Mathematica. However, I couldn't get a solution. It takes long to compute and doesn't compute it. How did you arrive at such an expression? I want to follow the same to derive the expression when there is no cosine term. | |
| Dec 4, 2022 at 1:56 | vote | accept | CfourPiO | ||
| Dec 4, 2022 at 1:55 | comment | added | CfourPiO | Thank you for the answer. Really appreciate it. I think I did some mistake arriving at this formula from my original physical problem. Indeed, $\beta$ term makes it a zero expectation and that’s not expected of the original physical problem. However, I was also looking for a solution when $\beta$ is not present. Thank you for this. I will again look into the physical problem. | |
| Dec 2, 2022 at 16:57 | history | answered | Carlo Beenakker | CC BY-SA 4.0 |