A few important integrals involving the product of two universal associated Legendre polynomials P_(l′)^(m′)(x),P_k^n′~′(x)and x^(2a)(1-x^2)^(-p-1),x^b(1±x)^(-p-1)and x^c(1-x^2)^(-p-1)(1±x)are evaluated ...A few important integrals involving the product of two universal associated Legendre polynomials P_(l′)^(m′)(x),P_k^n′~′(x)and x^(2a)(1-x^2)^(-p-1),x^b(1±x)^(-p-1)and x^c(1-x^2)^(-p-1)(1±x)are evaluated using the operator form of Taylor’s theorem and an integral over a single universal associated Legendre polynomial.These integrals are more general since the quantum numbers are unequal,i.e.l~′≠k~′and m~′≠n~′.Their selection rules are also given.We also verify the correctness of those integral formulas numerically.展开更多
文摘A few important integrals involving the product of two universal associated Legendre polynomials P_(l′)^(m′)(x),P_k^n′~′(x)and x^(2a)(1-x^2)^(-p-1),x^b(1±x)^(-p-1)and x^c(1-x^2)^(-p-1)(1±x)are evaluated using the operator form of Taylor’s theorem and an integral over a single universal associated Legendre polynomial.These integrals are more general since the quantum numbers are unequal,i.e.l~′≠k~′and m~′≠n~′.Their selection rules are also given.We also verify the correctness of those integral formulas numerically.
