arXiv subjects
Publications and source records attributed to Branko Saric.
Based on the total integrability we first define an integral of a real valued function f as an interval function associated to its antiderivative F. By introducing the concept of the residue of a function into the real analysis, the relationship between the so defined integral and the generalized Riemann integral is established.
Based on the kinetic energy theorem, as one of the fundamental theorems from the classical mechanics, throughout the first part of the article an attempt has been made to derive the mathematical model of a material point motion in the three-dimensional spatial subspace of the integral four-dimensional space-time continuum and in the field of action of an active force F. Accordingly, with a view to surmounting the singularity problem on the one hand, as well as the moving perihelion problem of the planets on the other, as two acutely vexed questions within Newton's gravity concept, the paper ends with modification of Newton's gravity concept itself.
In the main part of the paper, on the basis of contour integration of complex meromorphic functions whose singularities lie onto an integration contour, in the first step, a concept of improper integrals absolute existence of meromorphic functions, as more general one with respect to the concept of improper integrals convergence (existence), is introduced into analysis. In the second step, in the case when a modulus of complex parameter tends to infinity, an interval of improper integrals convergence of parametric meromorphic functions is defined. In accordance with this, it is shown that the class of real valued meromorphic functions, whose finitely many isolated singularities lie onto a real axis segment, may be expanded into Fourier trigonometric series, separately. At all points of the segment, at which the meromorphic functions are continuous ones, the Fourier trigonometric series is summable and its sum is equal to the function values at those points. Finally, that all is illustrated by two representative examples.
In the introduction part of this paper, first of all, the concept of absolute integral sum of complex function is defined, as more general one with respect to the concept of integral as well as of integral sum of "ordinary'' integral calculus. In the main part of the paper, on the basis of redefined concepts of a spatial derivative as well as of a residue of a complex function, the fundamental results of Cauchy's calculus of residues of analytic functions are generalized and the concept of total value of improper integral of complex function is defined. Finally, the results in the form of fundamental lemmas, which are more general with respect to the results of Jordan's fundamental lemmas of Cauchy's calculus of residues of analytic functions, are derived for the general class of scalar complex function whose defining domain is a complex plane.
Analogously to the concept of a curvature of curve and surface, in the differential geometry, in the main part of this paper the concept of the curvature of the hyper-dimensional vector spaces of Riemannian metric is generally defined. The defined concept of the curvature of Riemannian spaces of higher dimensions M: M>1, in the further text of the paper, is functional related to the fundamental parameters of an internal geometry of space, more exactly, to components of Riemann-Christoffel's tensor of curvature. At the end, analogously to the concept of lines of curvature in the differential geometry, the concept of sub-spaces of curvature of Riemannian hyper-dimensional vector spaces is also generally defined.