Some physical applications of generalized Lambert functions

Abstract

In this paper we review the physical applications of the generalized Lambert function recently defined by the first author. Among these applications we mention the eigenstate anomaly of the H + 2 ion, the two dimensional two-body problem in general relativity, the stability analysis of delay differential equations and water-wave applications. We also point out that the inverse Langevin function is nothing else but a specially parametrized generalized Lambert function.

On the generalization of the Lambert $W$ function

“…Physical applications of this generalizations can be found in [21] 3 The case of one upper and one lower parameter…”

Twisted Fermi surface of a thin-film Weyl semimetal

“…Penetration length ξ of the surface Fermi arc into the bulk Weyl semimetal, calculated via ξ=1/Im q from the solution of the Weiss equation(8), for the same parameters as figure 2. The penetration length diverges at k z =±1.475, according to equation(13). At this critical momentum the Fermi arc merges with the bulk Weyl cones.…”

Siewert solutions of transcendental equations, generalized Lambert functions and physical applications

“…where x (α) is replaced with the approximate solution (20), is fulfilled with a relative error less than 0.003, as we can see in the plot of Fig.2. So, we have the approximate relation:…”

“…Alternative ways of solving, exactly or approximately, this equation were presented in [34], where the exact solution is written in terms of a Lambert generalized function. The solution for the case h = 0 was written as a generalized Lambert function in [19], [20].…”