Lambert W function

In mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse relation of the function $f(w)=we^w$, where is any complex number and $e^w$ is the exponential function. The function is named after Johann Lambert, who considered a related problem in 1758. Building on Lambert's work, Leonhard Euler described the W function per se in 1783. Despite its early origins and wide use, its properties were not widely recognized until the 1990s thanks primarily to the work of Corless.

For each integer $k$ there is one branch, denoted by $W\_k\backslash left(z\backslash right)$, which is a complex-valued function of one complex argument. $W\_0$ is known as the principal branch. These functions have the following property: if $z$ and $w$ are any complex numbers, then : $w\; e^\{w\}\; =\; z$ holds if and only if : $w=W\_k(z)\; \backslash \; \backslash \; \backslash text\{\; for\; some\; integer\; \}\; k.$

When dealing with real numbers only, the two branches $W\_0$ and $W\_\{-1\}$ suffice: for real numbers $x$ and $y$ the equation : $y\; e^\{y\}\; =\; x$ can be solved for $y$ only if $x\backslash geq\backslash frac\{-1\}e$; yields $y=W\_0\backslash left(x\backslash right)$ if $x\backslash geq0$ and the two values $y=W\_0\backslash left(x\backslash right)$ and $y=W\_\{-1\}\backslash left(x\backslash right)$ if .

The Lambert W function's branches cannot be expressed in terms of elementary functions. It is useful in combinatorics, for instance, in the enumeration of trees. It can be used to solve various equations involving exponentials (e.g. the maxima of the Planck, Bose–Einstein, and Fermi–Dirac distributions) and also occurs in the solution of delay differential equations, such as $y\text{'}\backslash left(t\backslash right)=a\backslash \; y\backslash left(t-1\backslash right)$. In biochemistry, and in particular enzyme kinetics, an opened-form solution for the time-course kinetics analysis of Michaelis–Menten kinetics is described in terms of the Lambert W function.

Provided by Wikipedia