Approximate expressions for solutions to two kinds of transcendental equations with applications

Abstract

In a broad spectrum of physics and engineering applications, transcendental equations have to be solved in order to determine their roots. Exact and explicit algebraic expression of solutions to such equations is, in general, impossible. Analytical approximate solutions to two kinds of transcendental equations with wide applications are presented. These approximate root formulas are systematically established by using the Padé approximant and show high accuracy. As an application of the proposed approximations, a highly accurate expression of the effective mass of the spring for a spring-mass system is obtained. The method described in this paper is also applied to other transcendental equations in physics and engineering applications.

Transmission Line Circuit and Equation for an Electrolyte-Filled Pore of Finite Length

“…A Padé approximation of order [1/2] of the tan(nϑ − ) term in Eq. ( 12) yields the approximate solution [22]…”

An Approximate Solution to the Transcendental Equation Problem of the Finite Square Well Potential in Quantum Mechanics

“…In the subsequent section, this formal mathematical approach will be utilized to solve a specific class of transcendental equations encountered by scientists and engineers. We've chosen a couple of these equations, which can be found in many areas of physics in various forms, such as the study of the bound energy levels of a particle moving inside a symmetrical finite square well potential in quantum mechanics [5,[8][9][10][11][12][13][14].…”

“…Some approaches employ algebraic functions to approximate the various functions in the transcendental equation and therefore change the equation into an algebraic equation. In contrast, others use the Padé approximation to transform the equation into a rational one [5][6][7][8][9][10][11][12][13][14][15][16]. The primary objective of this study is to use a novel approach to obtain formal and analytical approximate solutions for such equations.…”

“…The primary objective of this study is to use a novel approach to obtain formal and analytical approximate solutions for such equations. To accomplish this, we have chosen the transcendental equation associated with the bound state of a particle moving in a symmetrical square well potential in quantum mechanics [5][6][7][8][9][10][11][12][13][14]. This problem was an elementary academic problem.…”