2022 A Generalised Continuous Primitive Integral and Some of its Applications
S. Mahanta, S. Ray
Author Affiliations +
Real Anal. Exchange 47(2): 297-322 (2022). DOI: 10.14321/realanalexch.47.2.1645501301


Using the Laplace derivative a Perron type integral, the Laplace integral, is defined. Moreover, it is shown that this integral includes Perron integral and to show that the inclusion is proper, an example of a function is constructed, which is Laplace integrable but not Perron integrable. Properties of integrals such as fundamental theorem of calculus, Hake's theorem, integration by parts, convergence theorems, mean value theorems, the integral remainder form of Taylor's theorem with an estimation of the remainder, are established. It turns out that concerning the Alexiewicz norm, the space of all Laplace integrable functions is incomplete and contains the set of all polynomials densely. Applications are shown to Poisson integral, a system of generalised ordinary differential equations and higher-order generalised ordinary differential equation.


Download Citation

S. Mahanta. S. Ray. "A Generalised Continuous Primitive Integral and Some of its Applications." Real Anal. Exchange 47 (2) 297 - 322, 2022. https://doi.org/10.14321/realanalexch.47.2.1645501301


Published: 2022
First available in Project Euclid: 10 February 2023

Digital Object Identifier: 10.14321/realanalexch.47.2.1645501301

Primary: 26A39 , 34A06
Secondary: 26A27

Keywords: Henstock integral , Laplace continuity , Laplace derivative , Laplace integral , ordinary differential equation , Perron integral , Poisson integral

Rights: Copyright © 2022 Michigan State University Press


This article is only available to subscribers.
It is not available for individual sale.

Vol.47 • No. 2 • 2022
Back to Top