Let
f
f
be a complex-valued locally integrable function on
[
0
,
+
∞
)
[0, + \infty )
, and let
L
f
Lf
be its Laplace transform, whenever and wherever it exists. We review some known methods, exact and approximate, for recovering
f
f
from
L
f
Lf
. Since numerical algorithms need auxiliary information about
f
f
near
+
∞
+ \infty
, we note that the behavior of
f
f
near
+
∞
+ \infty
depends on the behavior of
L
f
Lf
near 0 +, hence that our ability to retrieve
f
f
is limited by the class of momentless functions, namely, all functions
f
f
such that
L
f
(
s
)
Lf(s)
converges absolutely for
Re
(
s
)
>
0
\operatorname {Re} (s) > 0
and satisfies
\[
L
f
(
s
)
=
o
(
s
n
)
near
0
+
for
n
=
0
,
1
,
2
,
⋯
.
Lf(s) = o({s^n}){\text { near }}0 + \quad {\text {for}}\;n = 0,1,2, \cdots .
\]
We discuss the space
Z
Z
of momentless functions and complex distributions, then construct a family of elements in this space which defy various plausible conjectures.