Timeline for Computing the fractional Laplacian of power function
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 22, 2020 at 12:52 | vote | accept | Hiro | ||
| May 22, 2020 at 12:05 | comment | added | Mateusz Kwaśnicki | I noticed that Thm 3.1 is a wrong reference; should be Table 1. Once again sorry for all errors in the comments above. | |
| May 22, 2020 at 12:04 | answer | added | Mateusz Kwaśnicki | timeline score: 6 | |
| May 22, 2020 at 11:54 | comment | added | Mateusz Kwaśnicki | Sure, I will write it as an answer in a minute. Strange: the same rendering error happened here, but it disappeared when I refreshed the tab. | |
| May 22, 2020 at 11:50 | comment | added | Hiro | @MateuszKwaśnicki Thanks again. Oh, now I understand; in my browser the formula inside the comment is not properly displayed (I see it divided in two pieces with "whenever defined" in the middle). For the sake of clarity, could you please write it in an answer? | |
| May 22, 2020 at 11:47 | comment | added | Mateusz Kwaśnicki | Ah, I noticed a typo in my previous comment: should be $|x|^{p-\alpha}$, not $|x|^{p-n-\alpha}$, sorry. This is exactly as you say, with $q = p - \alpha$, but the constant depends on $n$, $\alpha$ and $p$. | |
| May 22, 2020 at 11:38 | comment | added | Hiro | @MateuszKwaśnicki Thanks! Could you clarify the $x$ dependence in your result? I expect it to be something like $-C_{N,\alpha} |x|^q$. | |
| May 22, 2020 at 11:32 | comment | added | Mateusz Kwaśnicki | This follows easily from the composition rule for the Riesz potential kernel. See Theorem 3.1 in my survey [Kwaśnicki, Fractional Laplace Operator and its Properties, in: A. Kochubei, Y. Luchko, Handbook of Fractional Calculus with Applications. Volume 1: Basic Theory, De Gruyter Reference, De Gruyter, Berlin, 2019], DOI:10.1515/9783110571622-007 for a rigorous statement and discussion. | |
| May 22, 2020 at 11:29 | comment | added | Mateusz Kwaśnicki | Yes: $(-\Delta)^{\alpha/2} [|x|^p] = 2^\alpha \Gamma((p+n)/2) \Gamma((\alpha-p)/2) (\Gamma((p+n-\alpha)/2) \Gamma((-p/2))^{-1} |x|^{p-n-\alpha}$ whenever defined. | |
| May 22, 2020 at 11:25 | history | asked | Hiro | CC BY-SA 4.0 |