Timeline for Integral involving Bessel and Laguerre function
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 10, 2022 at 11:58 | answer | added | Johannes Trost | timeline score: 2 | |
| Aug 9, 2022 at 11:24 | comment | added | Johannes Trost | That would mean that the integral is independent of the exponent $d$ of $r$ under the integral. | |
| Aug 9, 2022 at 11:11 | comment | added | Ryo Ken | I m sure of it@ Johannes Trost | |
| Aug 8, 2022 at 15:04 | comment | added | Johannes Trost | I try to check your result for the integral that you stated in the comments. I can not reproduce it, yet. Are you sure, that it is correct ? | |
| Aug 8, 2022 at 11:41 | comment | added | Ryo Ken | Thank you a lot@ Johannes Trost. I will use it. | |
| Aug 8, 2022 at 10:25 | comment | added | Johannes Trost | There IS in fact a simplification: for integer parameters the confluent hypergeometric function reduces of course considerably : $ _1F_1(n+m,n,z)=\exp(z) \sum_{j=0}^{m} {m \choose j } z^{j} / (n)_{j}$ (I have no link at hands for that formula, but should be online somewhere) | |
| Aug 7, 2022 at 13:15 | comment | added | Ryo Ken | Thank you a lot | |
| Aug 7, 2022 at 13:07 | comment | added | Johannes Trost | Not to my knowledge. | |
| Aug 7, 2022 at 12:26 | comment | added | Ryo Ken | Thnak you for your help. If we use your relations then the integral is equivalent to the sum $\sum^k_{j=0} \binom{k+1}{k-j}(-2)^j (j+1)_{1}F_{1}\left(j+2,2,-\frac{c^{2}}{4\ a}\right)$. Is there a closed formula for this | |
| Aug 7, 2022 at 12:06 | comment | added | Johannes Trost | The Laguerre polynomial can be found in NIST Special functions: dlmf.nist.gov/18.5.E12 . And use $\int^\infty_0 e^{-ar^2}J_1(cr)r^\delta dr =$ $\frac{1}{4} c a^{-1-\frac{\delta}{2}} \Gamma(1+\frac{\delta}{2}) _{1}F_{1}\left(1+\frac{\delta}{2},2,-\frac{c^{2}}{4\ a}\right)$. (Mathematica result) Could not find a reference for the integral, but maybe later. So called Confluent Hypergeometric function $_{1}F_{1}$ is here functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F1 | |
| S Aug 7, 2022 at 11:25 | review | First questions | |||
| Aug 7, 2022 at 17:29 | |||||
| S Aug 7, 2022 at 11:25 | history | asked | Ryo Ken | CC BY-SA 4.0 |