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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) = wew, where w is any complex number and ew 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 Wk(z), which is a complex-valued function of one complex argument. W0 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 {\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 W0 and W−1 suffice: for real numbers x and y the equation y e y = x {\displaystyle ye^{y}=x} can be solved for y only if x ≥ −1/e; gets y = W0(x) if x ≥ 0 and the two values y = W0(x) and y = W−1(x) if −1/e ≤ 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). 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 principal branch W0 is denoted Wp in the Digital Library of Mathematical Functions, and the branch W−1 is denoted Wm there. The notation convention chosen here (with W0 and 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 this: Since the inverse function of f(w) = ew is called the logarithm, it makes sense to call the inverse "function" of the product wew as "product logarithm". (Technical note: like the complex logarithm, it is multivalued and thus W is described as the converse relation rather than inverse function.) It is related to the omega constant, which is equal to W0(1). 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 wew. The equation Lambert considered was x = x m + q . {\displaystyle x=x^{m}+q.} Euler transformed this equation into the form x a − x b = ( a − b ) c x a + b . {\displaystyle x^{a}-x^{b}=(a-b)cx^{a+b}.} 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 ln ⁡ x = c x a . {\displaystyle \ln x=cx^{a}.} 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. Elementary properties, branches and range There are countably many branches of the W function, denoted by Wk(z), for integer k; W0(z) being the main (or principal) branch. W0(z) is defined for all complex numbers z while Wk(z) with k ≠ 0 is defined for all non-zero z. With W0(0) = 0 and limz→0 Wk(z) = −∞ for all k ≠ 0. The branch point for the principal branch is at z = −1/e, with a branch cut that extends to −∞ along the negative real axis. This branch cut separates the principal branch from the two branches W−1 and W1. In all branches Wk with k ≠ 0, there is a branch point at z = 0 and a branch cut along the entire negative real axis. The functions Wk(z), k ∈ Z are all injective and their ranges are disjoint. The range of the entire multivalued function W is the complex plane. The image of the real axis is the union of the real axis and the quadratrix of Hippias, the parametric curve w = −t cot t + it. Inverse The range plot above also delineates the regions in the complex plane where the simple inverse relationship W ( n , z e z ) = z {\displaystyle W(n,ze^{z})=z} is true. f = z e z {\displaystyle f=ze^{z}} implies that there exists an n .... 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