In mathematics, the Lambert W function, also called the Omega function or product logarithm, is a set of functions, namely the branches of the inverse relation of the function f(w) = we^{w} where e^{w} is the exponential function and w is any complex number. In other words, the defining equation for W(z) is
for any complex number z.
Since the function ƒ is not injective, the relation W is multivalued (except at 0). If we restrict attention to realvalued W then the relation is defined only for x ≥ −1/e, and is doublevalued on (−1/e, 0); the additional constraint W ≥ −1 defines a singlevalued function W_{0}(x). We have W_{0}(0) = 0 and W_{0}(−1/e) = −1. Meanwhile, the lower branch has W ≤ −1 and is denoted W_{−1}(x). It decreases from W_{−1}(−1/e) = −1 to W_{−1}(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 and also occurs in the solution of delay differential equations, such as y'(t) = a y(t − 1).
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Terminology
The Lambert Wfunction is named after Johann Heinrich Lambert. The main branch W_{0} is denoted by Wp in the Digital Library of Mathematical Functions and the branch W_{−1} is denoted by Wm there.
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