A trigonometric curve is a real plane curve where each coordinate is given parametri-cally by a truncated Fourier series. The trigonometric curves frequently arise in various areas of mathematics, physics, and engineering. Some trigonometric curves can be also represented implicitly by bivariate polynomial equations. In this paper, we give algorithms for (a) simplifying a given parametric repre...

Abstract It is well-known that the finite difference discretization of Laplacian eigenvalue problem ? ?u = ?u leads to a matrix (EVP) Ax ?x where A Toeplitz-plus-Hankel. Analytical solutions tridiagonal matrices with various boundary conditions are given in recent work Strang and MacNamara. We generalize results develop analytical certain generalized problems (GEVPs) ?Bx which arise from elemen...

Turetzkii [Uchenye Zapiski, Vypusk 1 (149) (1959), 31–55, (English translation in East J. Approx. 11 (2005) 337–359)] considered quadrature rules of interpolatory type with simple nodes, with maximal trigonometric degree of exactness. For that purpose Turetzkii made use of orthogonal trigonometric polynomials of semi–integer degree. Ghizzeti and Ossicini [Quadrature Formulae, Academie-Verlag, B...

Trigonometric functions are often needed in embedded real-time software. To fulfill concrete resource demands, different implementation strategies of trigonometric functions are possible. In this paper we analyze the resource demands of iterative calculations compared to other implementation strategies, using the trigonometric functions as a case study. By analyzing the worst-case execution tim...

We prove the case ρ = 4 of the following conjecture of Koumandos and Ruscheweyh: let s μ n (z) := ∑n k=0 (μ)k k! z k , and for ρ ∈ (0, 1] let μ(ρ) be the unique solution of ∫ (ρ+1)π 0 sin(t − ρπ)tμ−1dt = 0 in (0, 1]. Then we have | arg[(1− z)ρs n (z)]| ≤ ρπ/2 for 0 < μ ≤ μ(ρ), n ∈ N and z in the unit disk of C and μ(ρ) is the largest number with this property. For the proof of this other new re...

Starting from certain inequalities for hyperbolic and trigonometric functions, we obtain some general inequalities for functions and their inverses. As applications, refinements and new inequalities for hyperbolic and trigonometric functions are pointed out.

When we fix one side and draw different tangential quadrilaterals having the same side lengths but different angles we observe that their incenters lie on a circle. Based on a known formula expressing the incircle radius of a tangential quadrilateral by its tangent lengths, some older results will be presented in a new light and the equation of the before mentioned circle will appear. This circ...

In this note, we describe a method for establishing trigonometric inequalities that involve symmetric functions in the cosines of the angles of a triangle. The method is based on finding a complete set of relations that define the cosines of such angles.

This study is concerned with the visualization of monotone data using a piecewise C rational trigonometric interpolating scheme. Four positive shape parameters are incorporated in the structure of rational trigonometric spline. Conditions on two of these parameters are derived to attain the monotonicity of monotone data and other two are left free. Figures are used widely to exhibit that the pr...