The Lambert W function is a relatively young mathematical function with plenty of interesting applications. The algebraic equations leading to the Lambert W function are known since centuries, but the first treatise on the W function itself did not appear until 1996. After the seminal paper of R. M. Corless et al. [39], scientists have found surprisingly many applications of this function in the theory of projectile motion, wave physics, cell growth models, solar wind, enzyme kinetics, plasma and quantum physics, general relativity, prey and parasite models, twitter events, and so on. During the writing of the text, I kept in mind all of those scientists who are not mathematicians, but would like to better understand the W function, or a given aspect of it which is particularly necessary in their work. Therefore the text is written to be accessible for everyone with only basic knowledge in calculus and complex numbers. This is particularly apparent in Chapter 2: when introducing notions like branch cut, or Riemann surface we give full details for the more elementary and much better known smaller cousin of W, the logarithm function. I hope that this is a valid explanation for my mathematician colleagues who would otherwise find the lengthier introductions and explanations unnecessary.
It turned out soon after 1996 that more general equations are there which come from practical applications but are not solvable with the help of the W function. Motivated by this revelation, several directions of generalization of this function arose in the last couple of decades. Many aspects of these generalizations are also incorporated in this text.
This book is the very first one in the English language1 entirely dedicated to the Lambert W function, its generalizations, and its applications.
Our work intends to be a reference book which contains all the information one needs when trying to find a result on the Lambert function. In order to make the finding easier, the most important formulas and results are framed. But also, this is a book for an audience with wider interest: high school teachers can find motivation in Chapter 1 to examine equations not solvable by standard classroom methods; scientists from many fields may find interesting applications in the book; calculus professors can get ideas for teaching and practicing standard knowledge (derivation, integration, asymptotics, differential equations) on non-standard functions.
If a book targets a wider audience, it should be made clear what are the prerequisites. It depends on the given chapter. The first chapter needs basic real analysis, the second chapter requires some topology and complex function theory, and a very tiny bit of understanding group theory when the monodromy is discussed. The third chapter uses numerical approximations, matrix algebra, and computer programming. These cover the first part of the book. The second part does not need new knowledge, we discuss the generalized Lambert functions based on the tools we had developed in the first part. The third part is different: here we depart from mathematics, and discuss some applications of the W function and its generalizations. Here familiarity with Newtonian physics will be useful.
The researcher who is looking for interesting research problems can consult the Further Notes sections (especially in Chapter 2) in which questions are there for further research. The Further Notes sections contain not only research problems, but information for further studies on questions which, for some reason, are left out from this text. There are Mathematica codes on some pages of this book. Instead of typing these in, the reader can request them via e-mail: istvanmezo81@gmail.com.
In at least one aspect, however, the text is painfully incomplete. The generalizations of the Lambert function (the content of Part II) are just recently attracting attention from the researchers, and there are still many questions to work out; so, again, researchers looking for research problems, go ahead! There are many interesting questions here, and many papers to be written!
Certainly, it would have been impossible to put every application of W and its generalizations into this book. We selected the simpler and more accessible ones which might attract a wider audience. Such a selection is obviously subjective, but we hope that the reader will enjoy to learn about examples such as how the falling time of an object depends on the resistance of the air, or how the fuel consumption varies during the traveling of a rocket when the change in the mass caused by the same fuel consumption is taken into account.
From a pedagogical point of view, it is probably interesting to follow through the investigation of the properties of a function, which is arguably the simplest one outside the circle of the high school math class functions, such as log, sin and the others.
István Mező
Nanjing and Budapest, 2019–2021.