On Finite Integrals Involving Trigonometric, Bessel, and Legendre Functions

Richard L. Lewis
1969 Mathematics of Computation  
A finite integral involving the product of powers of trigonometric functions, up to two associated Legendre functions, and zero or one Bessel function is evaluated. When certain combinations of the otherwise complex function parameters are integers, the resulting expression becomes greatly simplified. So restricting the parameters, this still quite general case may be transformed into four canonical forms, each of which admits rapid convergence of the only nonterminating series in the
more » ... es in the expressions. Finally, closed form expressions are obtained for a number of special cases. ■ In some recent work by the author on boundary-value problems, it became necessary to evaluate finite integrals containing the products of associated Legendre functions in the integrand. Due to the large amplitude oscillations of these functions over the interval of integration, numerical quadrature techniques proved unreliable. A multiple series expression for the integral under consideration is presented, and then transforms of an otherwise rather untractable expression are introduced in order to obtain computable expressions when certain combinations of the parameters involved are integers. In general, however, the parameters (p, <r, v, b, ß, r¡, X, ß) have arbitrary complex values, provided they are so restricted as to allow the integral to exist. The integral considered then is
doi:10.2307/2004421 fatcat:g43xrq3zzvcpzatsbszmgb7may