Ring-Shaped Potential and a Class of Relevant Integrals Involved Universal Associated Legendre Polynomials with Complicated Arguments
We find that the solution of the polar angular differential equation can be written as the universal associated Legendre polynomials. Its generating function is applied to obtain an analytical result for a class of interesting integrals involving complicated argument, that is, ∫-11Pl′m′xt-1/1+t2-2xt...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2017-01-01
|
| Series: | Advances in High Energy Physics |
| Online Access: | http://dx.doi.org/10.1155/2017/7374256 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832545353291792384 |
|---|---|
| author | Wei Li Chang-Yuan Chen Shi-Hai Dong |
| author_facet | Wei Li Chang-Yuan Chen Shi-Hai Dong |
| author_sort | Wei Li |
| collection | DOAJ |
| description | We find that the solution of the polar angular differential equation can be written as the universal associated Legendre polynomials. Its generating function is applied to obtain an analytical result for a class of interesting integrals involving complicated argument, that is, ∫-11Pl′m′xt-1/1+t2-2xtPk′m′(x)/(1+t2-2tx)(l′+1)/2dx, where t∈(0,1). The present method can in principle be generalizable to the integrals involving other special functions. As an illustration we also study a typical Bessel integral with a complicated argument ∫0∞Jn(αx2+z2)/(x2+z2)nx2m+1dx. |
| format | Article |
| id | doaj-art-1de3c311f7764d229007c906d919bbc4 |
| institution | Kabale University |
| issn | 1687-7357 1687-7365 |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in High Energy Physics |
| spelling | doaj-art-1de3c311f7764d229007c906d919bbc42025-02-03T07:26:04ZengWileyAdvances in High Energy Physics1687-73571687-73652017-01-01201710.1155/2017/73742567374256Ring-Shaped Potential and a Class of Relevant Integrals Involved Universal Associated Legendre Polynomials with Complicated ArgumentsWei Li0Chang-Yuan Chen1Shi-Hai Dong2School of Science, Beijing University of Chemical Technology, Beijing 100029, ChinaNew Energy and Electronics, Yancheng Teachers University, Yancheng 224002, ChinaCIDETEC, Instituto Politécnico Nacional, Unidad Profesional ALM, 07700 Ciudad de México, MexicoWe find that the solution of the polar angular differential equation can be written as the universal associated Legendre polynomials. Its generating function is applied to obtain an analytical result for a class of interesting integrals involving complicated argument, that is, ∫-11Pl′m′xt-1/1+t2-2xtPk′m′(x)/(1+t2-2tx)(l′+1)/2dx, where t∈(0,1). The present method can in principle be generalizable to the integrals involving other special functions. As an illustration we also study a typical Bessel integral with a complicated argument ∫0∞Jn(αx2+z2)/(x2+z2)nx2m+1dx.http://dx.doi.org/10.1155/2017/7374256 |
| spellingShingle | Wei Li Chang-Yuan Chen Shi-Hai Dong Ring-Shaped Potential and a Class of Relevant Integrals Involved Universal Associated Legendre Polynomials with Complicated Arguments Advances in High Energy Physics |
| title | Ring-Shaped Potential and a Class of Relevant Integrals Involved Universal Associated Legendre Polynomials with Complicated Arguments |
| title_full | Ring-Shaped Potential and a Class of Relevant Integrals Involved Universal Associated Legendre Polynomials with Complicated Arguments |
| title_fullStr | Ring-Shaped Potential and a Class of Relevant Integrals Involved Universal Associated Legendre Polynomials with Complicated Arguments |
| title_full_unstemmed | Ring-Shaped Potential and a Class of Relevant Integrals Involved Universal Associated Legendre Polynomials with Complicated Arguments |
| title_short | Ring-Shaped Potential and a Class of Relevant Integrals Involved Universal Associated Legendre Polynomials with Complicated Arguments |
| title_sort | ring shaped potential and a class of relevant integrals involved universal associated legendre polynomials with complicated arguments |
| url | http://dx.doi.org/10.1155/2017/7374256 |
| work_keys_str_mv | AT weili ringshapedpotentialandaclassofrelevantintegralsinvolveduniversalassociatedlegendrepolynomialswithcomplicatedarguments AT changyuanchen ringshapedpotentialandaclassofrelevantintegralsinvolveduniversalassociatedlegendrepolynomialswithcomplicatedarguments AT shihaidong ringshapedpotentialandaclassofrelevantintegralsinvolveduniversalassociatedlegendrepolynomialswithcomplicatedarguments |