Lambert W Function

The Lambert W function, also called the omega function or the product logarithm, is defined as the inverse of the function that sends a number w to the product of w and the exponential of w, and because that function is not one to one over the whole real line, the Lambert W function is multivalued and is typically split into two real branches. It takes its name from the eighteenth century Swiss mathematician Johann Heinrich Lambert, who studied a related class of series, though the function in its modern form and notation, along with a systematic account of its properties, was established in a widely cited 1996 paper by Robert Corless, Gaston Gonnet, David Hare, David Jeffrey and Donald Knuth. The Lambert W function is the standard tool for solving equations in which an unknown quantity appears both inside and outside of an exponential term, and it arises naturally in combinatorics, in the analysis of algorithms, in the study of delay differential equations, and in several areas of physics and engineering.