The kernel of operators uncoupling systems of linear partial differential equations

The Laplace Transform exp[t(A-pD)]-exp[t(A 0 -pD)] where D is a diagonal matrix with diagonal elements d i , A is a constant n ×n matrix and A 0 , ij = A ij if d i =d j and A ij =0 otherwise, arises as the transform of the kernel of a Fredholm operator which maps the solutions of a system of linear partial differential equations which are uncoupled, onto those of a similar system but coupled by the matrix A. An inversion procedure is described and used to generate a series expression for the kernel.