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 ) = w e w {\displaystyle f(w)=we^{w}} , where w {\displaystyle w} is any complex number and e w {\displaystyle 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 {\displaystyle k} there is one branch, denoted by W k ( z ) {\displaystyle W_{k}\left(z\right)} , which is a complex-valued function of one complex argument. W 0 {\displaystyle W_{0}} is known as the principal branch. These functions have the following property: if z {\displaystyle z} and w {\displaystyle w} are any complex numbers, then
w e w = z {\displaystyle we^{w}=z}
holds if and only if
w = W k ( z ) for some integer k . {\displaystyle w=W_{k}(z)\ \ {\text{ for some integer }}k.}
When dealing with real numbers only, the two branches W 0 {\displaystyle W_{0}} and W − 1 {\displaystyle W_{-1}} suffice: for real numbers x {\displaystyle x} and y {\displaystyle y} the equation
y e y = x {\displaystyle ye^{y}=x}
can be solved for y {\displaystyle y} only if x ≥ − 1 e {\textstyle x\geq {\frac {-1}{e}}} ; yields y = W 0 ( x ) {\displaystyle y=W_{0}\left(x\right)} if x ≥ 0 {\displaystyle x\geq 0} and the two values y = W 0 ( x ) {\displaystyle y=W_{0}\left(x\right)} and y = W − 1 ( x ) {\displaystyle y=W_{-1}\left(x\right)} if − 1 e ≤ x < 0 {\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 ′ ( t ) = a y ( t − 1 ) {\displaystyle 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.
Terminology The notation convention chosen here (with W 0 {\displaystyle W_{0}} and W − 1 {\displaystyle W_{-1}} ) follows the canonical reference on the Lambert W function by Corless, Gonnet, Hare, Jeffrey and Knuth. The name "product logarithm" can be understood as follows: since the inverse function of f ( w ) = e w {\displaystyle f\left(w\right)=e^{w}} is termed the logarithm, it makes sense to call the inverse "function" of the product w e w {\displaystyle we^{w}} the "product logarithm". Like the complex logarithm, it is multivalued and thus W is described as a converse relation rather than inverse function. It is related to the omega constant, which is equal to W 0 ( 1 ) {\displaystyle W_{0}\left(1\right)} .
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