# Stieltjes transform

## Stieltjes transform

[′stēlt·yəs ‚tranz·fȯrm]
(mathematics)
A form of the Laplace transform of a function where the usual Riemann integral is replaced by a Stieltjes integral.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Since each [S.sub.n] is a Stieltjes transform of a positive measure (with total mass 3), the limit is of the form
We are therefore interested in [S.sub.(1)](z) since it gives the required Stieltjes transform
The Stieltjes transform of the asymptotic zero distribution is
Here, [z.sub.i] represents the LST (Laplace Stieltjes Transform), evaluated at point [[eta].sub.i], of the busy period in a M/M/1 queue with arrival rate [[lambda].sub.i] and service rate [[mu].sub.i].
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. The 1941 edition was published by Princeton University Press.
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.
(viii) f is the result of a Stieltjes transform i.e.
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.
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 [21]:
LOPEZ, Asymptotic expansions of generalized Stieltjes transforms of algebraically decaying functions, Stud.

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