Algebraic approximations for transcendental equations with applications in nanophysics

Semiconductor Quantum Wells with BenDaniel-Duke Boundary Conditions and Janus Nanorods

“…In several cases (see for instance [20]), the authors use Barker's formula for the energy levels in a square well [15]. Much more precise analytical expressions for these energy are available in the literature [8,9], for the case of constant mass; in this paper, we propose similar formulas, considering the case of position-dependent mass.…”

“…Approximate analytical solutions of these equations were obtained for deep wells p ≪ 1 ðÞ [15] and in the general case [8,9,16,17].…”

“…In its simplest form, for instance, in approximations like sin x ≃ x, for x ≪ 1, this "algebraization" is largely used. But what is really interesting is to use algebraic approximations of the trigonometric functions valid on their whole domain of definition, as de Alcantara Bonfin and Griffiths proposed in a recent paper [8]; such analytical approximations have been studied and extended by other authors [9].…”

“…(70) is a transcendental one, and its solutions cannot be expressed as a finite combination of elementary functions. A quite popular analytical approximation for the tangent function has been proposed by de Alcantara-Bonfim [8] and generalized by the present author [9]:…”

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

“…However, their results showed large errors. Based on the algebraic approximations of trigonometric functions, it is possible to transform a class of transcendental equations in approximate, tractable algebraic equations [4,11,12]. As the algebraization used in those papers is, to a certain extent, an ad hoc procedure, this approximation must be used with a certain caution in order to avoid the appearance of spurious roots or of roots with too large errors [12].…”