On the integration of products of Whittaker functions with respect to the second index

TL;DR

This paper develops new formulas for evaluating definite integrals involving the product of two Whittaker functions with respect to the second index, expanding the mathematical tools available for such integrals and their applications.

Contribution

It introduces novel integral formulas for Whittaker functions with respect to the second index, using complex contour integration and symmetry relations, complementing existing results.

Findings

Derived new integral formulas for Whittaker functions

Extended the range of integrals solvable with these functions

Discussed a physical application in radiative transport

Abstract

Several new formulas are developed that enable the evaluation of a family of definite integrals containing the product of two Whittaker W-functions. The integration is performed with respect to the second index, and the first index is permitted to have any complex value, within certain restrictions required for convergence. The method utilizes complex contour integration along with various symmetry relations satisfied by the Whittaker functions. The new results derived in this paper are complementary to the previously known integrals of products of Whittaker functions, which generally treat integration with respect to either the first index or the primary argument. A physical application involving radiative transport is discussed.