Highly accurate analytical approximate solutions for the transcendental equation y tan y = x with applications

Abstract

This technical note examines the function y(x), defined implicitly by the transcendental equation y tan y = x, which arises in various scientific and engineering applications. The function y(x) is a multivalued function, consisting of multiple branches. The Padé approximation method is applied to derive analytical approximations for each branch. The proposed approach requires the evaluation of only two square roots for the first branch and one square root for each subsequent branch. The precision of these explicit approximations can be further enhanced through the application of Schröder’s iteration, which entails computing only one additional tangent function. The resulting approximate expressions maintain high accuracy across both small and large values of x. Highly accurate analytical expressions for the buckling load of a uniform column with one end free and the other end elastically restrained under axial compressive loading, as well as for the analysis of the spring effective mass in a spring–mass vibration system, are derived using the proposed approximations.