Lambert W function

The product logarithm Lambert W function plotted in the complex plane from −2 − 2i to 2 + 2i

The graph of $y=W(x)$ for real $x<6$ and $y>-4$. The upper branch (blue) with $y\geq -1$ is the graph of the function $W_{0}$ (principal branch), the lower branch (magenta) with $y\leq -1$ is the graph of the function $W_{-1}$. The minimum value of $x$ is at $\left\{-1/e,-1\right\}$.

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 $w$ 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.

For each integer $k$ there is one branch, denoted by $W_{k}\left(z\right)$, 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

$we^{w}=z$

holds if and only if

$w=W_{k}(z)\ \ {\text{ for some integer }}k.$

When dealing with real numbers only, the two branches $W_{0}$ and $W_{-1}$ suffice: for real numbers $x$ and $y$ the equation

$ye^{y}=x$

can be solved for $y$ only if ${\textstyle x\geq {\frac {-1}{e}}}$; yields $y=W_{0}\left(x\right)$ if $x\geq 0$ and the two values $y=W_{0}\left(x\right)$ and $y=W_{-1}\left(x\right)$ if ${\textstyle {\frac {-1}{e}}\leq x<0}$.

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 $y'\left(t\right)=a\ y\left(t-1\right)$. 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.

Main branch of the Lambert W function in the complex plane, plotted with domain coloring. Note the branch cut along the negative real axis, ending at ${\textstyle -{\frac {1}{e}}}$.

The modulus of the principal branch of the Lambert W function, colored according to $\arg W\left(z\right)$