# Stieltjes Integral

## Stieltjes integral

[′stēlt·yəs ‚int·ə·grəl]
(mathematics)
The Stieltjes integral of a real function ƒ(x) relative to a real function g (x) of bounded variation on an interval [a,b ] is defined, analogously to the Riemann integral, as a limit of a sum of terms ƒ(ai ) [g (xi ) - g (xi-1)] taken as partitions of the interval shrink. Denoted Also known as Riemann-Stieltjes integral.

## Stieltjes Integral

a generalization of the definite integral proposed in 1894 by T. Stieltjes. In this generalization, the limit of the Riemann sums ∑f(ξi)(xi – xi-1) is replaced by the limit of the sums ∑f(ξi) [Φ(xi) – Φ(xi-1)]. where the integrating function ϕ(x) is a function of bounded variation (seeVARIATION OF A FUNCTION). If ϕ(x) is differentiable, then the Stieltjes integral can be expressed in terms of the Riemann integral (if it exists):

References in periodicals archive ?
In 2000, Dragomir (2) answered to the problem of approximating the Stieltjes integral [[integral].
From a different view point, the problem of approximating the Stieltjes integral [[integral].
Remark Using the Stieltjes integral by Dragomir (2), generalization of the Ostrowski problem (3) was considered, so our results are natural to generalize some results obtained by Barnett et al.
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.
and integrating by parts in the Stieltjes integral [[integral].
This follows by the fact that, in this case [PSI] becomes absolutely continuous on [a, b], the Stieltjes integral [[integral].
Jankowski, Monotone iterative method to second order differential equations with deviating arguments involving Stieltjes integral boundary conditions, Dynam.
n[member of]N] of functions inside a Stieltjes integral.
Jankowski, Positive solutions for second order impulsive differential equations involving Stieltjes integral conditions, Nonlinear Anal.
It is important to indicate that boundary conditions involving Stieltjes integrals appeared in some papers in which the problem of existence of positive solutions to differential equations have been discussed.
mu]] is to express these quantities as Stieltjes integrals.
15), respectively, in terms of Stieltjes integrals.

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