Laplace contour integrals and linear differential equations

TL;DR

This paper investigates the properties of Laplace contour integrals that solve specific linear differential equations, focusing on growth, asymptotics, zero distribution, and solutions, to deepen understanding of their analytical behavior.

Contribution

It characterizes the main properties of Laplace contour integrals associated with linear differential equations, including growth, asymptotics, and zero distribution, advancing the theoretical understanding.

Findings

Determines growth order and asymptotic behavior of solutions.

Analyzes zero distribution and existence of special solutions.

Explores Nevanlinna functions related to these integrals.

Abstract

The purpose of this paper is to determine the main properties of Laplace contour integrals $$\Lambda(z)=\frac1{2\pi i}\int_\CC\phi_L(t)e^{-zt}\,dt,$$ that solve linear differential equations $$L[w](z):=w^{(n)}+\sum_{j=0}^{n-1}(a_j+b_jz)w^{(j)}=0.$$ This concerns, in particular, the order of growth, asymptotic expansions, the Phragm\'en-Lindel\"of indicator, the distribution of zeros, the existence of sub-normal and polynomial solutions, and the corresponding Nevanlinna functions.