72 - Integral Representations: Laplace's Method*

Abstract

This chapter focuses on the Laplace's method for integral representations. This method is applicable to linear ordinary differential equations. It yields an integral representation of the solution. The idea behind the method is that the solution of a linear ordinary differential equation can be written as a contour integral. The procedure considers Lz[·] a linear differential operator with respect to z, and it is supposed that the ordinary differential equation to be solved has the form Lz[u(z)] = 0. The procedure then looks for a solution of Lz[u(z)] = 0 in the form u(z)= ∫K(z,ɛ)v(ɛ)dɛ for some function v(ɛ), and some contour C in the complex ɛ plane. The function K(z,ɛ) is called the kernel. The chapter highlights that for the case where Lz[·] is a linear operator with polynomial coefficients, the solution is easy to find using the Laplace kernel.