[Paper Review] Extension of Algebraic Solutions Using The Lambert W Function

Sidney Edwards|arXiv (Cornell University)|Feb 24, 2019

Sports Dynamics and Biomechanics9 references4 citations

TL;DR

This paper extends algebraic solutions to exponential and logarithmic equations using the Lambert W function, deriving inversion formulas and polar coordinate representations through identities and tetration. It provides exact analytical solutions for complex equations—such as those modeling film thickness in chemical engineering—by leveraging W function branches and transformations, enabling closed-form solutions where traditional methods fail.

ABSTRACT

The Lambert W function has utility for solving various exponential and logarithmic equations arranged in the form of $g(x)e^{g(x)}$. Using the Lambert W function and tetration, a variety of categorized inversion formulas are presented. Related techniques are then used to derive polar forms of exponential and related functions.

Motivation & Objective

• To generalize solutions to exponential and logarithmic equations beyond elementary functions using the Lambert W function.
• To derive inversion formulas and identities for equations of the form $ f(x)e^{g(x)} $ using the W function.
• To apply the W function to represent solutions in polar coordinates and relate them to rotations and transformations.
• To connect the W function with tetration and iterated exponentials, enabling solutions to power tower and root problems.
• To provide exact analytical solutions for real-world equations in science and engineering, such as film thickness dynamics.

Proposed method

• Derives inversion formulas for equations like $ y = x e^{a x} $ by manipulating them into the standard $ z = W(z) e^{W(z)} $ form.
• Applies the W function's multivalued nature, particularly the $ W_0 $ and $ W_{-1} $ branches, to solve equations with multiple real solutions.
• Uses tetration identities and the inverse of $ x^x $ to define the tetra square root via $ x = e^{W(\ln y)} $, extending to higher-order roots.
• Transforms Cartesian equations into polar forms using trigonometric substitutions and applies the W function to solve for $ r $ and $ \theta $.
• Derives change-of-base formulas for the W function across different exponent bases, enabling broader applicability.
• Applies the W function to solve systems involving linear combinations of exponentials and trigonometric functions in polar coordinates.

Experimental results

Research questions

• RQ1 How can the Lambert W function be systematically applied to solve a broad class of exponential and logarithmic equations?
• RQ2 What are the exact analytical solutions for equations involving $ f(x)e^{g(x)} $ forms, particularly in applied contexts like chemical engineering?
• RQ3 How can the W function be used to represent solutions in polar coordinates, and what geometric interpretations arise from such representations?
• RQ4 What is the connection between the Lambert W function and tetration, and how can it be used to define inverse operations like the tetra root?
• RQ5 Can the W function’s multivalued branches be combined meaningfully to recover all real solutions of a given equation?

Key findings

• The paper derives an exact solution for film thickness dynamics in chemical engineering: $ D(t) = \frac{b}{a}\left[1 + W(-e^{-1 - a^2 t / b^2})\right] $, valid for $ t \geq 0 $.
• It establishes that the inverse of $ x^x $ is $ e^{W(\ln y)} $, providing a closed-form expression for the tetra square root.
• The solution for $ y = x e^{a x} $ is $ x = \frac{W(a y)}{a} $, demonstrating a general method for solving such transcendental equations.
• The paper derives a polar form solution for $ y = \frac{1}{B} \ln\left(\frac{x}{A}\right) $, expressing $ y $ as a function of $ x $ using $ W $-based transformations.
• It shows that $ e^{B a u + B b v} = \frac{b}{A B a} W\left(\frac{A B a}{b} e^{\frac{B(a^2 + b^2)}{b} u}\right) $, enabling solution of linear-exponential systems.
• Through trigonometric substitution, the method yields a polar representation: $ y = \frac{\sec \phi}{B} \ln\left( \frac{\tan \phi}{A B} W( A B \cot \phi \, e^{B \csc \phi \, x} ) \right) - \tan \phi \, x $, valid for specific parameter ranges.