2021 CAP Virtual Congress / Congrès virtuel de l'ACP 2021

The offset logarithm function and some applications in physics: A generalization of the Lambert W function

9 Jun 2021, 15:53

3m

Underline Conference System

Oral (Non-Student) / Orale (non-étudiant(e)) Theoretical Physics / Physique théorique (DTP-DPT) W3-2 Mathematical and Theoretical Physics (DTP) / Physique mathématique et physique théorique (DPT)

Speaker

Aude Maignan (university Grenoble Alpes)

Description

We describe the offset logarithm function and illustrate its applications to physics. The offset log can be considered an extension of the logarithm function, and it can also be considered as a generalization of the Lambert W function.
The offset log function shows up in a variety of physical and biological models, such as the mean field model (Weiss model) of ferromagnetism, and models of carbon nanoribbons. It provides a convenient framework for describing phenomena in which the behaviour of a physical variable at one location or time, has an exponential or logarithmic relationship to the value of attribute at a displaced location or time.

The offset logarithm function of order $k$ is the multivalued function $w=L_k(z)$ of the complex variable $z$, which satisfies the equation
$$\frac{w-k}{w+k}{e}^w=z(1)$$
Each of $k$, $z$, and $w$ can be in general a complex number. However, for this, we consider $k$, $z$ and $w$ as real numbers. Approximate solutions to Equation (1) can be calculated numerically, but a study of the analytic properties of that equation can provide insight into the system being described.

The offset log function is multi-branch. Each branch of $L_k(z)$ is an analytic function for certain regions of $k$ and $z$ values. We present the main characteristics of the $L_k(z)$ function. For each domain of the $k$ parameter, the properties of $L_k$ in terms of the branches are given. We show that the number of solutions of Equation (1) depends upon the domains of $k$ and $z$. For these domains we give the number of solutions and we give the interval where each solution lies. We identify the cases where a solution is a multiple solution. We give a symbolic formula for the multiple solutions.