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2000 Math. Subject Classification: 33E12, 65D20, 33F05, 30E15The paper deals with analysis of several techniques and methods for the
numerical evaluation of the Wright function. Even if the focus is mainly on
the real arguments’ values, the methods introduced here can be used in the
complex plane, too. The approaches presented in the paper include integral
representations of the Wright function, its asymptotic expansions and
summation of series. Because the Wright function depends on two parameters
...
Zeta-generalized-Euler-constant functions,
and
defined on the closed interval [0, ∞), where γ(1) is the Euler-Mascheroni constant and
(1) = ln
, are studied and estimated with high accuracy.
Accurate estimates of real Pochhammer products, lower (falling) and upper (rising), are presented. Double inequalities comparing the Pochhammer products with powers are given. Several examples showing how to use the established approximations are stated.
We obtain some approximate identities whose accuracy depends on the bottom of the discrete spectrum of the Laplace-Beltrami operator in the automorphic setting and on the symmetries of the corresponding Maass wave forms. From the geometric point of view, the underlying Riemann surfaces are classical modular curves and Shimura curves.
The paper focuses on a low-rank tensor structured representation of Slater-type and Hydrogen-like orbital basis functions that can be used in electronic structure calculations. Standard packages use the Gaussian-type basis functions which allow us to analytically evaluate the necessary integrals. Slater-type and Hydrogen-like orbital functions are physically more appropriate, but they are not analytically integrable. A numerical integration is too expensive when using the standard discretization...
Values of the Epstein zeta function of a positive definite matrix and the knowledge of matrices with minimal values of the Epstein zeta function are important in various mathematical disciplines. Analytic expressions for the matrix theta functions of integral matrices can be used for evaluation of the Epstein zeta function of matrices. As an example, principal coefficients in asymptotic expansions of variance of the lattice point count in the random ball are calculated for some lattices.
Four formulas of the Menzerath–Altmann law are tested from the point of view of their applicability and suitability. The accuracy of related approximations of measured data is examined by the least square method at first. Then the accuracy of calculated parameters in the formulas under consideration is compared statistically. The influence of neglecting parameter is investigated as well. Finally, the obtained results are discussed by means of an illustrative example from quantitative linguistics....
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