Laplace contour integrals and linear differential equations

Abstract

The purpose of this paper is to determine the main properties of Laplace contour integrals $$Λ(z)=\frac1{2πi}\int_\CCϕ_L(t)e^{-zt}\,dt,$$ that solve linear differential equations $$L[w](z):=w^{(n)}+\sum _{j=0}^{n-1}(a_j+b_{j}z)w^{(j)}=0.$$ 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.