A Pad\'e approximant approach to two kinds of transcendental equations with applications in physics

TL;DR

This paper introduces a Padé approximant method to analytically solve two types of transcendental equations in physics, providing accurate, practical formulas suitable for undergraduate teaching and applications like diffraction and quantum potential problems.

Contribution

The paper presents a novel application of Padé approximants combined with Lagrange inversion to solve transcendental equations in physics educational contexts.

Findings

Approximate formulas achieve high accuracy in physics problems.

Method applicable to various textbook examples.

Enhances pedagogical approaches in physics education.

Abstract

In this paper, we obtain the analytical solutions of two kinds of transcendental equations with numerous applications in college physics by means of Lagrange inversion theorem, and rewrite them in the form of ratio of rational polynomials by second order Pad\'e approximant afterwards from a practical and instructional perspective. Our method is illustrated in a pedagogical manner for the purpose that students at the undergraduate level will be beneficial. The approximate formulas introduced in the paper can be applied to abundant examples in physics textbooks, such as Fraunhofer single slit diffraction, Wien's displacement law and Schr\"odinger equation with single or double $\delta$ potential. These formulas, consequently, can reach considerable accuracies according to the numerical results, therefore they promise to act as valuable ingredients in the standard teaching curriculum.