Complex Path Integral Representation for Semiclassical Linear Functionals Original Research Article

Semiclassical linear functionals are characterized by the distributional equationD(φL)+ψL=0 whereφandψare arbitrary polynomials with the condition deg(ψ)⩾1. Two cases are considered: (A) deg(φ)>deg(ψ) (B) deg(φ)⩽deg(ψ). In an earlier work by the authors (J. Comput. Appl. Math.57(1995), 239–249) integral representations are given for semiclassical functionals in case (A). Here the problem is continued and case (B) is solved: it is always possible to choose some pathγin the complex plane such that every solution, regular or not, ofD(φL)+ψL=0 can be represented in the form ⦠L, p⦔=∫γ w(z) p(z) dzwherew(z) is a solution of the differential equation (φw)′+ψw=0. In some cases, the expression forLis a singular integral and a regularization process is given.