Representation of the Stieltjes Integral in Terms of the Riemann Integral Depending on a Parameter

V. A. Chernyatin

University of Szczecin

Abstract: In this paper, we obtain a new formula for the representation of the Riemann–Stieltjes integral of a continuous function in terms of the passage to the limit with respect to the parameter in a Riemann integral depending on this parameter. The derivation of this formula is based on the study of the functional properties of the solution of the auxiliary difference equation of first order representing the weighted first difference of a given function in the form of a simple first difference of an unknown function. The result obtained can be used for the analytic and approximate calculation of Stieltjes integrals.

UDC: 517.3

Received: 23.06.2003
Revised: 30.08.2004

DOI: 10.4213/mzm2651

 English version: Mathematical Notes, 2005, 78:6, 853–866

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