APPLICATION OF THE LAMBERT FUNCTIONS IN SOLVING TRANSCENDENTAL EQUATIONS

Abstract

The Lambert W function: $w=W(z)$ is defined as the solution of the equation $we^{w} = z$ where e is the base of the natural logarithm. There are several equations of the form $e^{x}\left(\frac{a_1x+a_2}{a_3x+a_4}\right)=a_5$, $e^x(a_1+a_2\sqrt x)=a_3$, $e^x\left(\frac{a_1+a_2\sqrt x}{a_3+a_4\sqrt x}\right)=a_5$, e.t.c, which can be solved using corresponding generalized Lambert W functions. These equations occur in different fields in math and engineering, such as Magnetic Micro-Electro-Mechanical-Structures, Chemical Engineering, and other fields. This project shows solutions to these equations, the branch structure of the Lambert functions, expanding solutions into a series, and analyzing the radius of convergence.