Uncoupling the Differential Equations Arising from a Technique for EvaluatingIndefinite Integrals Containing Special Functions or Their Products

摘要

A previous article by Piquette and Van Buren described an analytical techniquefor evaluating indefinite integrals involving special functions or their products. The technique replaces the integral by an inhomogeneous set of coupled first-order differential equations. This coupled set does not explicitly contain the special functions of the integrand, and any particular solution of the set is sufficient to obtain an analytical expression for the indefinite integral. It is shown here that the coupled set arising from the method always occurs in normal form. Hence, it is amenable to the method of Forsyth (6) for uncoupling such a set. That is, the solution of the set can be made to depend upon the solution of a single differential equation of order equal to the number of equations in the set. Any particular solution of this single equation is then sufficient to yield the desired indefinite integral. As examples, the uncoupled equation is given here for integrals involving (i) the product of two Bessel functions, (ii) the product of Hermite functions, or (iii) the product of two Laguerre functions, and a tabulation of integrals of these types is provided. The method can be used to extend the integration capabilities of symbolic-mathematics computer programs so that they can handle broad classes of indefinite integrals containing special functions or their products. Keywords: Reprints. (kr)