Excel-oriented calculator for finding the values of some special functions of mathematical and theoretical physics

Abstract

The goal of the work. Development of an EXCEL-oriented calculator for finding the values of special functions of mathematical and theoretical physics. Research subject: special functions of mathematical and theoretical physics. Research method: numerical methods for obtaining values of special functions. The obtained results. EXCEL-oriented algorithms for calculating special functions are proposed: for direct and inverse hyperbolic functions, Bessel functions, complete and incomplete gamma functions, polygamma functions, complete and incomplete beta functions and their partial derivatives, Fresnel integrals, integral circular and hyperbolic trigonometric functions, integral exponent, integral logarithm, dilogarithm, Clausen function, and associated Clausen function, Lobachevsky function, integral secant function (Sievert function). All proposed algorithms use only numerical integration operations (rectangle method) and calculations of finite sums of functional series. The ways for adaptively determining the number of intervals during numerical integration that necessary to achieve the accepted calculation accuracy are described. For improper integrals, a procedure for replacing them with ordinary definite integrals that ensure calculations with a given accuracy is proposed. The similar procedures for determining the number of terms in the n-th partial sum of a functional series with an infinite number of terms are proposed. It is shown that the proposed algorithms provide calculation accuracy no less than the Mathcad package. An algorithm for calculating function values using Chebyshev polynomials is proposed and implemented. The algorithm can be used to calculate the values of an arbitrary continuous function specified by the user.