Sixty Minute Talks

Ronald E. Mickens, Atlanta, Georgia, USA

Application of the Lambert W Function to NSFD Numerical Schemes for Differential Equations
The Lambert W function is used to construct finite difference discretizations for both ordinary and partial differential equations. These schemes have the feature that a number of important mathematical properties of the original differential equations are exactly incorporated into the corresponding discrete models. This presentation will first provide a summary of the obtained results, followed by a general introduction to the NSFD scheme methodology [1] and its application to problems in which the Lambert W function arises. Several possible extensions of this work will also be discussed .

[1] Ronald E. Mickens, Nonstandard Finite Difference Models of Differential Equations (World Scientific, Singapore, 1994 / eBook, May 10, 2014).

Jon Borwein, CARMA, Department of Mathematics, University of Newcastle

The Lambert W function in Optimisation and Analysis
The Lambert W function is the functional inverse, appropriately defined, of x exp(x). This function is not elementary but perhaps it should be. It was first used by Johann Heinrich Lambert (1728-1777) but was only named twenty years ago [1]. It is now implemented in computer packages such as Maple, Mathematica and SAGE. So your computer probably knows more about W than you do. Until a function is named it cannot gather a literature. For example, [1] now has over 3684 Google Scholar citations. I will describe recent work illustrating on W's remarkable utility in convex analysis and in optimisation.

This is joint work with Scott Lindstrom [2].

[1] R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, and D.E. Knuth, “On the Lambert W Function,” Advances in Computational Mathematics, 5 (1996), 329-359. [2] Jonathan Borwein and Scott Lindstrom, “The Lambert W function in Optimization,” to appear in PAFA. Available at https://www.carma.newcastle.edu.au/jon/lambert.pdf.

Gilbert Labelle, LaCIM, UQAM, Montreal

Special series related to Lambert W
A PDF file of the abstract is available here

John P. Boyd, Department of Atmos. Oceanic Space Science, University of Michigan

Forty Years of Adventures with Pseudospectral Methods, Chebyshev Proxy Rootfinding, the Bratu Equation and the Lambert W-function and Its Generalizations
The Bratu equation, $\Delta u + \lambda \exp(u)=0$, has several connections to the W-function. (i) A one-point pseudospectral apporoximation, either one-dimensional or two-dimensional, is moderately accurate for small and moderate amplitude with $A=\max(u)$ given without further approximation as a W-function of the parameter $\lambda$ (ii) A one-basis-function Galerkin approximation to the ODE solution is significantly more accurate than the pseudospectral approximation; $A(\lambda)$ is a generalization of the Lambert function which is however well approximated by W. For all these cases, the two real branches of W simultaneously capture both the upper and the lower branches of the Bratu solutions. The Lambert function has also been a challenging but illuminating case study for various forms of both direct and inverse Chebyshev polynomial approximation. Hermite-Pade approximants can capture two branches simultaneously and generalize Pade approximants by defining the approximation $f_{N/M}$ to be the solution of a quadratic equation. The coefficients of the quadratic are polynomials chosen so that $f_{[N/M]}$ approximates $f(x)$ in the appropriate sense. This ``Chebyshev-Shafer" strategy applies to multibranched functions in general and yields an exponential rate of convergence. The difficulties of unbounded interval and unbounded range are shown to be easily defeated by standard Chebyshev technology. Chebyshev approximations to functions of the unknown, rather than the parameter, ``polynomialize'' the transcendental equation to an easily-solved polynomial equation. This is the ``Chebyshev-Proxy Rootfinding" (CPR)Method of the author's recent book. We show how to overcome the difficulties of unbounded range and domain and fold point for Chebyshev proxies. Lastly, we offer some new globally accurate analytical approximations which are Never-Failing-Newton's-Initializations (NFNI) for the numerical computation of the W-function to near-machine epsilon through a ocde only ten lines long without loops or conditionals. Some larger themes on psi-series and implicit versus explicit representations of a function are also mentioned.

Vladimir Vinogradov, Department of Mathematics, Ohio University

Approximation of an integral transform of the Wright function in terms of the Lambert W function arising in probability theory
We discovered that the asymptotics of an integral transform of a particular Wright function in the neighborhood of the left endpoint of its parameter domain is given in terms of the principal real branch of the Lambert W function. This analysis result emerged in our studies on properties of specific exponential dispersion models whose members are used for modelling count data, and is equivalent to pointwise convergence of their variance functions characterized by the same value of an invariant of Esscher transform. Our “probabilistic” proof of this analysis result involves the Wright-function representation for the solution to a specific equation established in our article in press, and is closely related to recent work by Burridge, Kuznetsov, Kwaśnicki and Kyprianou on fluctuation properties of Lévy processes. We confirmed our asymptotic result numerically and welcome suggestions on its prospective purely analytic proof.

This is joint work with Richard B. Paris (University of Abertay Dundee)

Alexander Kheyfits, Department of Mathematics, CUNY

Some new problems involving Lambert W
Abstract not available.

Frank Garvan, Department of Mathematics, University of Florida

Lambert and Ramanujan
We consider Lambert and generalized Lambert series studied by Ramanujan. We start with some weighted partition identities and divisor functions. Along the way we meet Ramanujan's mock theta functions and tau function. We show how we used MAPLE to discover, prove and check results. In particular we introduce a new MAPLE package, thetaids, for proving theta function identities. The results for the tau function are joint with Michael Schlosser.

Erik Postma, Mathematics, Maplesoft

Finding Special Function Identities
The computer algebra system Maple has a programmatically accessible database of interrelated special function identities and conversion rules. This can be used for simplification and reformulation of expressions. While Maple's `simplify' command makes some use of this knowledge, it is not exhaustive. Identities hitherto unknown to the system may be programmatically generated, as we will see by example. We will also discuss the question of how to augment the database, in part by programmatic search and computation.

Thirty Minute Talks (Alphabetical by Last Name)

Samir Hamdi, Department of Civil and Water Engineering, Laval University

Explicit analytical solutions for critical and normal depths in circular channels based on Lambert W function
The critical and normal depths are quantities of fundamental importance in the analysis, design, operation and maintenance of open channels. This important hydraulic structures are often used in drainage, irrigation and water supply systems. The critical and normal depths are also significant parameters for understanding the flow characteristics in open channels. The governing equation for the critical and normal depths in circular channels are implicit and transcendental nonlinear algebraic equation. Their exact closed form solutions are not available. In this research work, new explicit analytical formulas for critical and normal depths are obtained without using empirical or curve fitting techniques. The solutions are derived by performing a series expansion of the logarithm of the governing equations and solving the simplified truncated equations analytically using symbolic computation. The resulting analytical solutions are based on Lambert function. It is shown that the Lambert solutions are more accurate and mathematically more rigorous than regression based curve fitting techniques. They are practical and simple for manual calculations or numerical modelling applications in circular open channel flows. The approach that we devised, for finding analytical and explicit Lambert function solutions, is simple and general and can be easily extended for finding closed-form formulae of critical and normal depths for a wide class of cross sections in open channel flows.

German Kalugin, Department of Physics, The Royal Military College of Canada

Properties of an asymptotic series for the Lambert W Function
The Lambert W function admits a number of asymptotic series, with coefficients known in terms of different special numbers. One such series was found by de Bruijn in 1961 (before W was named). I consider a series from 1995, with coefficients given in terms of 2-associated Stirling numbers. Unlike many asymptotic series, it has a domain of convergence. We present results in both the real and the complex case, by converting the series to a power series, and using the Implicit Function Theorem and the Weierstrass preparation theorem. We obtain an analytical description of the boundary of the convergence domain (it is not just a boring circle!). We also derived asymptotic expressions for the expansion coefficients using Darboux's theorem about expansions at algebraic singularities. The coefficients are found to exhibit asymptotically oscillating behavior.

This is joint work with David Jeffrey.

André LeClair, Department of Physics, Cornell University,

Transcendental equations satisfied by the individual zeros of Riemann $\zeta$, Dirichlet and modular L-functions
A PDF file of the abstract is available here

Ken Roberts, Department of Physics and Astronomy, Western University,

Solar Cells and the Lambert W Function
A PDF file of the abstract is available here

Fei Wang, Department of Applied Mathematics, Western University,

The series solution of Euler 's symmetric form of Lambert's trinomial equation
In 1758, Lambert solved the trinomial equation x = q + x^m. Later Euler transformed it into a symmetric form: x^a - x^b = (a -b) v x^{a+b} and also a series solution of this equation. However, the proof in Euler's original paper was not complete. The original paper was written in Latin with German and French translations in some universities in Europe. We will review Euler's proof and provide a new proof using the Cauchy integral representation, the residue theorem and a theorem about interpolation from Labelle's work in 1980.