#### Date of Award

2011

#### Degree Type

Thesis

#### Degree Name

Doctor of Philosophy

#### Program

Applied Mathematics

#### Supervisor

Dr. David Jeffrey

#### Abstract

This research studies analytical properties of one of the special functions, the Lambert W function. W function was re-discovered and included into the library of the computer-algebra system Maple in 1980’s. Interest to the function nowadays is due to the fact that it has many applications in a wide variety of fields of science and engineering.

The project can be broken into four parts. In the first part we scrutinize a convergence of some previously known asymptotic series for the Lambert W function using an experimental approach followed by analytic investigation. Particularly, we have established the domain of convergence in real and complex cases, given a comparative analysis of the series properties and found asymptotic estimates for the expansion coefficients. The main analytical tools used herein are Implicit Function Theorem, Lagrange Inversion Theorem and Darboux’s Theorem.

In the second part we consider an opportunity to improve convergence prop erties of the series under study in terms of the domain of,convergence and rate of convergence. For this purpose we have studied a new invariant transformation defined by parameter p, which retains the basic series structure. An effect of parameter p on a size of the domain of convergence and rate of convergence of the series has been studied theoretically and numerically using M a p l e . We have found that an increase in parameter p results in an extension of the domain of convergence while the rate of convergence can be either raised or lowered.

We also considered an expansion of W(x) in powers of Inx. For this series we found three new forms for a representation of the expansion coefficients in terms of different special numbers and accordingly have obtained different ways ito'compute the expansion coefficients. As an extra consequence we have obtained some combinatorial relations including the Carlitz-Riordan identities.

In the third part we study the properties of the polynomials appearing in the expressions for the higher derivatives of the Lambert W function. It is shown that the polynomial coefficients form a positive sequence that is log-concave and unimodal, which implies that the positive real branch of the Lambert W function is Bernstein and its derivative is a Stieltjes function.

In the fourth part we show that many functions containing Ware Stieltjes functions. In terms of the result obtained in the third part, we, in fact, obtain one more way to establish that the derivative of W function is a Stieltjes function. We have extended the properties of the set of Stieltjes functions and also proved a generalization of a conjecture of Jackson, Procacci &; Sokal. In addition, we have considered a relation of W to the class of completely monotonic functions and shown that W is a complete Bernstein function. .

We give explicit Stieltjes representations of functions of W, We also present integral representations of W which are associated with the properties of its being a Bernstein and Pick function. Representations based on Poisson and Burniston- Siewert integrals are given as well. The results are obtained relying on the fact that the all of the above mentioned classes are characterized by their own integral forms and using Cauchy Integral Formula, Stieltjes-Perron Inversion Formula and properties of W itself.

#### Recommended Citation

Kalugin, German Alekseevich, "Analytical properties of the Lambert W function" (2011). *Digitized Theses*. 3272.

https://ir.lib.uwo.ca/digitizedtheses/3272