Laplace contour integrals and linear differential equations

Abstract

The purpose of this paper is to determine the main properties of Laplace contour integrals Λ(z)=12πi∫Cϕ(t)e−ztdt

that solve linear differential equations L[w](z):=w(n)+∑j=0n−1(aj+bjz)w(j)=0.

This concerns, in particular, the order of growth, asymptotic expansions, the Phragmén–Lindelöf indicator, the distribution of zeros, the existence of sub-normal and polynomial solutions, and the corresponding Nevanlinna functions.