History and some modern applications of Lambert W function

ABSTRACT In 1737 Euler proved the irrationality of the number e, and the same property of \pi was proved by Johann Heinrich Lambert (1728 - 1777) in 1768. Lambert suspected that both \pi and e are transcendental but could not prove it. Transcendence of e was not established until 1873 by Hermite and then, based on a valuable hint from Hermite (some thirty pages long!), Carl Louis Ferdinand Lindemann showed that \pi is also a transcendental number. A real number that is a root of a polynomial equation with integer coefficients is considered an algebraic number, while real numbers that do not have this property are called transcendental. In the pertinent proofs either polynomials or a linear combination of the exponential functions \sum A_i exp (a_i) were used. Example of such expression is the famous Euler formula exp(i\pi)+exp(0)=1. Expressions of this type are also commonly used in Statistical Thermodynamics, in which Jaynes' formalism is applied to arrive at the Maxwell-Boltzmann distributions. Lambert introduced mixed expressions that combined the power terms and the exponential terms appearing as products. The simplest form of such mixed expression, Wexp(W)= x, defines implicitly the Lambert W function, W=W(x). Some curious properties of the transcendental number \omega=W(1), which is a remote cousin of the golden ratio, are discussed. Certain applications of the W function in solving problems in projectile motion and in Orbital Mechanics are shown. Finally, occurrence of the transitions from orderly motion in the gravitational field to a chaotic motion is illustrated by means of fractals and their attractors.