On the asymptotic behaviour of Laplace-type multiple integral solutions of linear differential equations

Multiple integral solutions of two classes of nth-order differential equations are considered. These representations have been known since the 19th century. This note points out the connection between these integrals and recent work of the first author and of Paris and Liakhovetski (Frac. Calc. Appl. Anal. 3(1) (2000) 63). The asymptotic behaviour is first obtained by rewriting these multiple integrals as Mellin–Barnes integrals. We then show how such integrals may be transformed into a generalised Faxén integral and an expansion of wider validity obtained.