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

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.