On Totalization of the $H_1$-Integral

Abstract

Based on the total H 1 -integrability concept, which is established in this paper, we shall try to show that at any point of a compact interval (a, b] in R, at which a point function F has no a discontinuity, F is the total H 1 -indefinite integral of a function dF ex being the limit of ∆F ex (I), where I ⊆ [a, b], on [a, b], without additional hypotheses on F . A residue function of F is introduced. The paper ends with a few of examples that illustrate the theory.

Fourier series of functions with infinite discontinuities

“…It is an old result (see [9]) that if γ : [−π, π] → R is a point function defined by γ (x, t) = +∞ k=1 Γ k (x, t) + (x − t) /2, where Γ k (x, t) = sin [k (x − t)] /k, for every fixed t ∈ (0, π), then the dispersion of function values on [−π, π] is as follows…”

“…The following theorem gives us the opportunity to compute the Fourier coefficients for a function that can take not only finite but infinite values within [−π, π], using the H 1 -integral, [9].…”

“…Perhaps this can be best exemplified by using the so-called total value of the generalized Riemann integrals introduced by Saric in his works [6,7,8,9]. This brand new theory of integration, which takes the notion of residues of real valued functions into account, gives us the opportunity to integrate real valued functions that are not integrable in any of the known integration methods until now.…”

On the HN-integration of spatial (integral) derivatives of multivector fields with singularities in RN

“…Macdonald [11] used the regular partition integral to overcome the deficiency in Hestenes' proof of Stokes' theorem, [1,6]. Sarić [15] defined a new integral named total H 1 -integral. This integral solves the problem in formulating the fundamental theorem of calculus in R whenever a primitive F is defined at the end points of [a, b] ⊂ R. Accordingly, in what follows, we will try to extend Cauchy's integral formula to anN-dimensional manifold M immersed in R N , for a large scale class of multivector fields F, and in the spirit of Hestenes' appealing proof.…”