TY - JOUR
T1 - Ring-Shaped Potential and a Class of Relevant Integrals Involved Universal Associated Legendre Polynomials with Complicated Arguments
AU - Li, Wei
AU - Chen, Chang Yuan
AU - Dong, Shi Hai
N1 - Publisher Copyright:
Copyright © 2017 Wei Li et al.
PY - 2017
Y1 - 2017
N2 - 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, ∫ - 1 1 P l ′ m ′ x t - 1 / 1 + t 2 - 2 x t P k ′ m ′ (x) / (1 + t 2 - 2 t x) (l ′ + 1) / 2 d x, 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 ∞ J n (α x 2 + z 2) / (x 2 + z 2) n x 2 m + 1 d x.
AB - 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, ∫ - 1 1 P l ′ m ′ x t - 1 / 1 + t 2 - 2 x t P k ′ m ′ (x) / (1 + t 2 - 2 t x) (l ′ + 1) / 2 d x, 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 ∞ J n (α x 2 + z 2) / (x 2 + z 2) n x 2 m + 1 d x.
UR - http://www.scopus.com/inward/record.url?scp=85015794980&partnerID=8YFLogxK
U2 - 10.1155/2017/7374256
DO - 10.1155/2017/7374256
M3 - Artículo
SN - 1687-7357
VL - 2017
JO - Advances in High Energy Physics
JF - Advances in High Energy Physics
M1 - 7374256
ER -