Two new trigonometric formulas with applications

We derive two new trigonometric identities and the corresponding Chebyshev identities. We also obtain new Fibonacci and Lucas Identities.     

Quadrature and Runge-Kutta formulas

Error estimation and step size adjustment procedures for many high order Runge-Kutta methods fail dramatically when applied to an equation of the form y'(x)=f(x). A practical remedy is proposed and its use exemplified with the Fehlberg [7,8] formulas. The management of storage for this important example is also discussed.     

Quadrature formulas for periodic functions and their application to the boundary element method

Two-dimensional and axisymmetric boundary value problems for the Laplace equation in a domain bounded by a closed smooth contour are considered. The problems are reduced to integral equations with a periodic singular kernel, where the period is equal to the length of the contour. Taking into account the periodicity property, high-order accurate quadrature formulas are applied to the integral operator. As a result, the integral equations are reduced to a system of linear algebraic equations. This substantially simplifies the numerical schemes for solving boundary value problems and considerably improves the accuracy of approximation of the integral operator. The boundaries are specified by analytic functions, and the remainder of the quadrature formulas decreases faster than any power of the integration step size. The examples include the two-dimensional potential inviscid circulation flow past a single blade or a grid of blades; the axisymmetric flow past a torus; and free-surface flow problems, such as wave breakdown, standing waves, and the development of Rayleigh-Taylor instability.     

Quadrature formulas for the Laplace and Mellin transforms

A discrete Laplace transform and its inversion formula are obtained by using a quadrature of the integral Fourier transform which is given in terms of Hermite polynomials and its zeros. This approach yields a convergent discrete formula for the two-sided Laplace transform if the function to be transformed falls off rapidly to zero and satisfies given conditions of integrability, achieving convergence also for singular functions. The inversion formula becomes a quadrature formula for the Bromwich integral. The use of asymptotic formulae yields an algorithm to compute the discrete Laplace transform by using only exponentials.     

Quadrature formulas for the generalized Riemann-Stieltjes integral

In this paper we propose a technique of approximation for the generalized Riemann-Stieltjes integral and we found an analogue for Newton-Cotes formulas in the case n = 2 and n = 3. *Beneficiary of a Socrates fellowship at the Department of Mathematics, University of Study of Cagliari, Via Ospedale, n. 72, Cagliari, 09124, Italy, in the period February – July 2002.     

Theta-function identities and the explicit formulas for theta-function and their applications

We define two quotients of theta-functions depending on two positive real parameters. We then show how they are connected with two parameters of Dedekind eta-function and the Ramanujan-Weber class invariants. Explicit formulas for determining values of the theta-function is derived, and several examples will be given and using them, we give some complete explicit results for the complete elliptic integral of the first kind and the Gaussian hypergeometric function. Also several new modular equations for the theta-function are derived.     

On integral formulas and their applications in functional calculus

Integral formulas from the theory of holomorphic functions of several variables are applied to explicit treatment of the analytic functional calculus for commuting operators in Banach spaces (commutative Banach algebras). Among others the integral formula of Henkin in smooth convex domains, is used. An application is also given to commuting normal operators in Hilbert spaces.     

Inequalities for trigonometric sums and applications

Aequationes mathematicae - We present various new inequalities for cosine and sine sums. Among others, we prove that 0.1$$\begin{aligned} 0\le \sum _{k=0}^n \frac{ (a)_{2k}}{(2k)!} \frac{\cos...     

Quadrature formulas on the unit circle based on rational functions

Quadrature formulas on the unit circle were introduced by Jones in 1989. On the other hand, Bultheel also considered such quadratures by giving results concerning error and convergence. In other recent papers, a more general situation was studied by the authors involving orthogonal rational functions on the unit circle which generalize the well-known Szeg polynomials. In this paper, these quadratures are again analyzed and results about convergence given. Furthermore, an application to the Poisson integral is also made.     

Quadrature formulas for classes with constraints on the Fourier coefficients

Translated from Matematicheskie Zametki, Vol. 49, No. 2, pp. 144–147, February, 1991.     

Quadrature formulae with multiple nodes and a maximal trigonometric degree of exactness

In this paper we consider interpolatory quadrature formulae with multiple nodes that have the maximal trigonometric degree of exactness. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quadrature formulae with multiple nodes and a maximal trigonometric degree of exactness

In this paper we consider interpolatory quadrature formulae with multiple nodes, which have the maximal trigonometric degree of exactness. Our approach is based on a procedure given by Ghizzeti and Ossicini (Quadrature formulae, Academie-Verlag, Berlin, 1970). We introduce and consider the so-called σ-orthogonal trigonometric polynomials of semi-integer degree and give a numerical method for their construction. Also, some numerical examples are included. The authors were supported in part by the Serbian Ministry of Science and Technological Development (Project: Orthogonal Systems and Applications, grant number #144004) and the Swiss National Science Foundation (SCOPES Joint Research Project No. IB7320-111079 “New Methods for Quadrature”).