AbstractThe 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.