Special Function System Research Articles

Calculation of special functions arising in the problem of diffraction by a dielectric ball

To apply the incomplete Galerkin method to the problem of the scattering of electromagnetic waves by lenses, it is necessary to study the differential equations for the field amplitudes. These equations belong to the class of linear ordinary differential equations with Fuchsian singularities and, in the case of the Lneburg lens, are integrated in special functions of mathematical physics, namely, the Whittaker and Heun functions. The Maple computer algebra system has tools for working with Whittaker and Heun functions, but in some cases this system gives very large values for these functions, and their plots contain various kinds of artifacts. Therefore, the results of calculations in the Maple11 and Maple2019 systems of special functions related to the problem of scattering by a Lneburg lens need additional verification. For this purpose, an algorithm for finding solutions to linear ordinary differential equations with Fuchsian singular points by the method of Frobenius series was implemented, designed as a software package Fucsh for Sage. The problem of scattering by a Lneburg lens is used as a test case. The calculation results are compared with similar results obtained in different versions of CAS Maple. Fuchs for Sage allows computing solutions to other linear differential equations that cannot be expressed in terms of known special functions.

СРЕДНЕКВАДРАТИЧЕСКОЕ ПРИБЛИЖЕНИЕ ФУНКЦИЙ РЯДАМИ ФУРЬЕ–БЕССЕЛЯ И ЗНАЧЕНИЯ ПОПЕРЕЧНИКОВ НЕКОТОРЫХ ФУНКЦИОНАЛЬНЫХ КЛАССОВ

It is known that many of the problems of mathematical physics, reduced to a differential equation with partial derivatives written in cylindrical and spherical coordinates, by using method of separation of variables, in particular, leads to the Bessel differential equation and Bessel functions. In practice, especially in problems of electrodynamics, celestial mechanics and modern applied mathematics most commonly used Fourier series in orthogonal systems of special functions. Given this, it is required to determine the conditions of expansion of functions in series into these special functions, forming in a given interval a complete orthogonal system. The work is devoted to obtaining accurate estimates of convergence rate of Fourier series by Bessel system of functions for some classes of functions in a Hilbert space L 2 := L 2 ([0 , 1] , x dx ) of square summable functions f : [0 , 1] → R with the weight x . The exact inequalities of Jackson–Stechkin type on the sets of L2 ( r ) 2 ( D ) , linking En − 1 ( f ) 2 — the best approximation of function f by partial sums of order n − 1 of the Fourier–Bessel series with the averaged positive weight of generalized modulus of continuity of m order Ω m (

A design and application of compound multi-functional sensor in wood-based panel processing

A compound multi-functional sensor was designed by the study on the on-line testing technology of wood-based panels, and its properties of shape, functions, size, resistance to special environment were studied in details. The operational principles of different sensors, technical flow of manufacturing, development of software systems of special functions, and the assessments of technical specification were also be introduced. This sensor adopted many new technologies, such as the applications of piezoresistant effect and heat sensitive effect can effectively measure the pressure and temperature, digital signal processing technology was used to extract and treat signals, and resist interference, encapsulation technology was used to keep the normal run of sensor under a harsh environment. Thus, the on-line compound multi-functional temperature/pressure sensor can be applied better to supervise the production of wood-based panels. All technical specifications of the compound multi-functional sensor were tested and the results met the requirements of the equipments.

Solution properties of the de Branges differential recurrence equation§

In his 1984 proof of the Bieberbach and Milin conjectures de Branges used a positivity result of special functions which follows from an identity about Jacobi polynomial sums that was published by Askey and Gasper in 1976 (Askey, R. and Gasper, G., 1976, Positive Jacobi polynomial sums II. American Journal of Mathematics, 98, 709–737.). The de Branges functions are defined as the solutions of a system of differential recurrence equations with suitably given initial values. The essential fact used in the proof of the Bieberbach and Milin conjectures is the statement . In 1991 Weinstein presented another proof of the Bieberbach and Milin conjectures, also using a special function system which (by Todorov and Wilf) was realized to be directly connected with de Branges’, , and the positivity results in both proofs are essentially the same. In this article we study differential recurrence equations equivalent to de Branges’ original ones and show that many solutions of these differential recurrence equations don’t change sign so that the above inequality is not as surprising as expected. Furthermore, we present a multiparameterized hypergeometric family of solutions of the de Branges differential recurrence equations showing that solutions are not rare at all.

Positivity and monotony properties of the de Branges functions

In his 1984 proof of the Bieberbach and Milin conjectures de Branges used a positivity result of special functions τkn(t) which follows from an identity about Jacobi polynomial sums that was published by Askey and Gasper in 1976.In 1991 Weinstein presented another proof of the Bieberbach and Milin conjectures, also using a special function system Λkn(t) which (by Todorov and Wilf) was realized to be directly connected with de Branges', τ˙kn(t)=-kΛkn(t), and the positivity results in both proofs τ˙kn(t)≤0 are essentially the same.By the relation τ˙kn(t)≤0, the de Branges functions τkn(t) are monotonic, and τkn(t)≥0 follows. In this article, we reconsider the de Branges and Weinstein functions, find more relations connecting them with each other, and make the above positivity and monotony result more precise, e.g., by showing τkn(t)≥(n−k+1)e−kt.

On the approximate solution of some two‐dimensional singular integral equations

AbstractAn approximation method for a wide class of two‐dimensional integral equations is proposed. The method is based on using a special function system. Orthonormality and good interaction with fundamental integral operators arising in partial differential equations are remarkable properties of this system. In addition, all the basis elements can easily be calculated by recurrence relations. Taking into account these properties we construct a numerical algorithm which does not require additional effort (such as quadrature) to compute the values of the fundamental operators on the basis elements. Copyright © 2001 John Wiley & Sons, Ltd.

Weinstein's functions and the askey-gasper identity

In his 1984 proof of the Bieberbach and Milin conjectures de Branges used a positivity result of spacial functions which follows from an identity about Jacobi polynomial sums that was found by Askey and Gasper in 1973, published in 1976. In 1991 Weinstein presented another proof of the Bieberbach and Milin conjectures, also using a special function system which (by Todorov and Wilf) was realized to be the same as de Branges'. In this article, we show how a variant of the Askey-Gasper identity can be deduced by a straight-forward examination of Weinstein's functions which are intimately related with a Löwner chain of the Koebe fucntion, and therefore with univalent functions.

On the de branges theorem

Recently, Todorov and Wilf independently realized thai de Branges original proof of the Bieberbach and Milin conjectures and the proof that was later given by Weinstein deal with the same special function system that de Branges had introduced in his work. In this article, we present an elementary proof of this statement based on the defining differential equations system rather than the closed representation of de Branges function system. Our proof does neither use special functions (like Wilf's) nor the residue theorem (like Todorov s) nor the closed representation (like both), but is purely algebraic. On the other hand, by a similar algebraic treatment the closed representation of de Branges function system is derived. Our whole contribution can be looked at as the study of properties of the Koebe function. Therefore, in a very elementary manner it is shown that the known proofs of the Bieberbach and Milin conjectures can be understood as a consequence of the Lowner differential equation, plus propertie...

On a complete orthonormal system of special functions

New complete orthonormal (CON) systems of special functions in the space of functions with summable squares are introduced.

Untersuchungen zur rekursion von l. de branges

The recursion of de Branges in the form , given in [3] and [4], is considered. In the first part, starting with arbitrary initial values τ k (0)K = 1, 2, …n, a connection between this recursion and the ordinary differential equations is established. Furthermore a conjunction of our method with that of Henrici [4] is derived. The second part is based on the initial values of de Brange, τ k (0) = n- k+1k = 1, 2, …n. Virtually by means of the Laplace transformation the special function system of de Branges is derived in a very easy manner.