Lambert W function
History

Lambert first considered the related Lambert's Transcendental Equation in 1758, which led to an article by Leonhard Euler in 1783 that discussed the special case of we.

The equation Lambert considered was

Euler transformed this equation into the form

Both authors derived a series solution for their equations.

Once Euler had solved this equation, he considered the case a = b {\displaystyle a=b}. Taking limits, he derived the equation

He then put a = 1 {\displaystyle a=1} and obtained a convergent series solution for the resulting equation, expressing x {\displaystyle x} in terms of c {\displaystyle c}.

After taking derivatives with respect to x {\displaystyle x} and some manipulation, the standard form of the Lambert function is obtained.

In 1993, it was reported that the Lambert W {\displaystyle W} function provides an exact solution to the quantum-mechanical double-well Dirac delta function model for equal charges —a fundamental problem in physics. Prompted by this, Rob Corless and developers of the Maple computer algebra system realized that "the Lambert W function has been widely used in many fields, but because of differing notation and the absence of a standard name, awareness of the function was not as high as it should have been."

Another example where this function is found is in Michaelis–Menten kinetics.

Although it was widely believed that the Lambert W {\displaystyle W} function cannot be expressed in terms of elementary (Liouvillian) functions, the first published proof did not appear until 2008.
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