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

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 I x and K x with respect to the index . The new family of definite integrals is useful in a variety of contexts in mathematical physics. In particular, the integral involving K x 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 generalize previously known results. An application to time-dependent radiation transport theory is also discussed. © 2009 American Institute of Physics. doi:10.1063/1.3265924