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

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.