Laplace Contour Integrals and Linear Differential Equations

The purpose of this paper is to determine the main properties of Laplace contour integrals $$\begin{aligned} \Lambda (z)=\frac{1}{2\pi i}\int _\mathfrak {C}\phi (t)e^{-zt}\,dt \end{aligned}$$ Λ ( z ) = 1 2 π i ∫ C ϕ ( t ) e - z t d t that solve linear differential equations $$\begin{aligned} L[w](z):=w^{(n)}+\sum _{j=0}^{n-1}(a_j+b_jz)w^{(j)}=0. \end{aligned}$$ L [ w ] ( z ) : = w ( n ) + ∑ 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.