More on the complex branches of the Lambert W function

The paper discusses the basic and advanced properties of the Lambert W function in the complex domain. Despite many works devoted to this classical topic (dealing mainly with the principal branch of the function), several important questions have not been discussed or answered yet. Therefore, the aim of this paper is twofold. First, it describes the real and imaginary parts of all the complex branches of the Lambert W function via certain implicit equations (in the real domain). Second, applying such implicit characterizations strengthens and extends the existing knowledge on the key topics of the Lambert W function theory, such as the distribution and localization of its function values. Our findings in both research directions also offer a new insight into another important topic, namely, approximations of the Lambert W function values.