Abstract:The method of brackets is an efficient method for the evaluation of a large class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the concepts of null and divergent series. These are formal representations of functions, whose coefficients a n have meromorphic representations for n 2 C, but might vanish or blow up when n 2 N. These ideas are illustrated with the evaluation of a variety of entries from the classical table of integrals by Gradshteyn and Ryzhik.
Abstract. Expectation values of powers of the radial coordinate in arbitrary hydrogen states are given, in the quantum case, by an integral involving the associated Laguerre function. The method of brackets is used to evaluate the integral in closed-form and to produce an expression for this average value as a finite sum.
Abstract. We present proofs for typical entries from the Gradshteyn-Ryzhik Table of Integrals using the Mellin transform method and computer algebra algorithms based on WZ theory. After representing an identity from the Table in terms of multiple contour integrals of Barnes' type and nested sums, we use Wegschaider's summation algorithm to find recurrences satisfied by both sides of this identity and check finitely many initial values.
Abstract:The method of brackets is a collection of heuristic rules, some of which have being made rigorous, that provide a flexible, direct method for the evaluation of definite integrals. The present work uses this method to establish classical formulas due to Frullani which provide values of a specific family of integrals. Some generalizations are established.
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