The inverse Laplace transform of the modified Lommel functions

摘要

Particular solutions of the inhomogeneous Bessel differential equation are the Lommel functions, usually denoted as s_(μ, ν)(x), S_(μ, ν)(x), and S_(μ, ν)(x). The inhomogeneous modified Bessel differential equation has the particular solutions t_(μ, ν)(x)=-i~(1-μ)s_(μ, ν)(ix), T_(μ, ν)(x)=-i~(1-μ)S_(μ, ν)(ix), and T_(μ, ν)(x)=-i~(1-μ)S_(μ, ν)(ix). For the modified Lommel functions T_(μ, ν)(x) and T_(μ, ν)(x) as well as T_(μ, ν)(√x) and T_(μ, ν)(√x), we give the inverse Laplace transform for μ=-1 and μ=0. For μ<-1, the inverse Laplace transform can be obtained from a recurrence relation.