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Concept
Lambert W function
Summary
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.
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
holds if and only if
When dealing with real numbers only, the two branches W0 and W−1 suffice: for real numbers x and y the equation
can be solved for y only if x ≥ −1/e; we get 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 relation 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.
The Lambert W function is named after Johann Heinrich Lambert. 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.
Official source
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