Cauchy’s residue theorem for a class of real valued functions

Abstract

Let [a, b] be an interval in Ê and let F be a real valued function defined at the endpoints of [a, b] and with a certain number of discontinuities within [a, b]. Assuming F to be differentiable on a set [a, b] \ E to the derivative f , where E is a subset of [a, b] at whose points F can take values ±∞ or not be defined at all, we adopt the convention that F and f are equal to 0 at all points of E and show that KH-vt b a f = F (b) − F (a), where KH-vt denotes the total value of the Kurzweil-Henstock integral. The paper ends with a few examples that illustrate the theory.

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

“…The obtained result provides an extension of Cauchy's integral formula from the calculus of residues in M (compare with results in [7,16]).…”

Fourier series of functions with infinite discontinuities

“…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.…”