Infinite integrals of Whittaker and Bessel functions with respect to their indices

Abstract

We obtain several new closed-form expressions for the evaluation of a family of infinite-domain integrals of the Whittaker functions W,x and M,x and the modified Bessel functions Ix and Kx with respect to the index . The new family of definite integrals is useful in a variety of contexts in mathematical phys- ics. In particular, the integral involving Kx represents a new example of the Kontorovich-Lebedev transform. We discuss the relationship between the results derived here and the previously known integrals of Whittaker and Bessel functions. In some cases, we obtain entirely new expressions, and in other cases, we gener- alize previously known results. An application to time-dependent radiation trans- port theory is also discussed. © 2009 American Institute of Physics.