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.

In the introductory part of this paper the so-called general Bromwich-Wagner's theorem has been generalized. Throughout the main part of the paper, an attempt has been made to derive conditions for convergence of the fundamental matrix of linear time-invariant timedelayed singular systems. The paper ends with a counterexample pointing out the fact that certain results of the theory of linear time-invariant singular systems are incorrect.2000 Mathematics Subject Classification: 93C05, 34A30, 34D05.1. Introduction. The introductory part is provided to give the generalization of the so-called general Bromwich-Wagner's theorem from Cauchy's calculus of residues, which has proven to be a powerful tool in the Laplace transforms theory. The main part is concerned with the theory of singular systems. On the basis of the result worded in the form of the theorem at the very beginning, throughout the main part of the paper, an attempt has been made to derive conditions for convergence of the fundamental matrix of linear time-invariant time-delayed singular systems. The paper ends with a counterexample pointing out the fact that certain results of the theory of linear timeinvariant singular systems, as for example a condition for the existence of infinite frequency dynamic modes (of impulsive modes) in a solution formulation of linear time-invariant singular systems, are incorrect.In the complex functions theory, more precisely in the theory of Cauchy's calculus of residues, the following result, which is well known as the general Bromwich-Wagner's theorem (see [4, Theorem 1, page 215]), is a fundamental result from the viewpoint of determining the function t f (t) whose Laplace transform is z F(z).

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.

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