The text covers the Stieltjes integral, fundamental formulas, the moment problem, absolutely and completely monotonic functions, Tauberian theorems, the bilateral Laplace transform, inversion and representation problems for the Laplace transform, and the Stieltjes transform
Our results hold in particular for functions which arise as the result of a Stieltjes transform and thus for certain rational functions and for the inverse square root.
The functions considered in (ii) to (vii) are all particular Stieltjes transforms, i.
The fact that we are also in the presence of Stieltjes transforms in cases (v) and (vi) has been observed in , the case (vii) was treated in .
Let us remark that the set of Stieltjes transforms is a subset of the set of completely monotone functions.
A function f on (0, [infinity]) is called a Stieltjes transform
if it can be written in the form
Except for the minus sign in front of s; the right-hand side is the Stieltjes transform
of a nonnegative measure.
Taking Laplace Stieltjes transforms
in equation (10) and solving for [Mathematical Expression Omitted], we get
We consider here the important example of the Stieltjes transforms :
LOPEZ, Asymptotic expansions of generalized Stieltjes transforms of algebraically decaying functions, Stud.
WONG, Explicit error terms for asymptotic expansions of Stieltjes transforms, J.