In this article we define the Riemann indefinite integral of functions of real variable and prove the linearity of that [1]. And we give some examples of the indefinite integral of some elementary functions. Furthermore, also the theorem about integral operation and uniform convergent sequence of functions is proved. provide the terminology and notation for this paper.

in the present paper for a large family of topological semigroups, namely foundation semigroups, for which topological groups and discrete semigroups are elementary examples, it is shown that ?(s) is the dual of a function algebra.

A common practice for computing an elementary transcendental function nowadays has two phases: reductions of input arguments to fall into a tiny interval and polynomial approximations for the function within the interval. Typically the interval is made tiny enough so that one won’t have to go for polynomials of very high degrees for accurate approximations. Often approximating polynomials as su...

Many of us who teach calculus and mathematical topics that use calculus have taken for granted that hand-held calculators use Taylor series or a variant to compute transcendental functions. Thus, it was a surprise to learn that this was not the case. The CORDIC method (Coordinate Rotation Digital Computer) was developed by Jack Volder [6] in the late 1950’s. Hewlett-Packard was quick to realize...

The Madelung constant-essentially the Coulomb energy density of a crystal-is usually calculated via Ewald error function expansions or, for the simpler cubic structures, by the 'cosech' series of modern vintage. By considering generalised functional equations for multidimensional zeta functions, we provide explicit expansions for the spatial potential and energy density of three-dimensional per...

Piece-wise first- and second-order approximations are employed to design commonly used elementary function generators for neural-network emulators. Three novel schemes are proposed for the first-order approximations. The first scheme requires one multiplication, one addition, and a 28-byte lookup table. The second scheme requires one addition, a 14-byte lookup table, and no multiplication. The ...

Elementary algebraic constraints on the form of an autocorrelation function C(t(w)+tau,t(w)) rule out some two-time scalings found in the literature as possible long-time asymptotic forms. The same argument leads to the realization that two usual definitions of many-time scale relaxation for aging systems are not equivalent.