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Lambert W function
The Lambert W function is a collection of functions, which form the inverse of the function f (z) = e z. Here e z is the complex exponential function and z is a complex number. For each complex number z the following applies
z = W (z) eW (z)
Explanation
The Lambert W function cannot be expressed in terms of elementary functions. It can be used to solve equations containing exponents (e.g. the maxima of the Planck, Bose-Einstein, and Fermi-Dirac distributions) and also occurs when solving derivatives, such as y' (t) = a y (t − 1).
Graph of W0(x) for −1/e ≤ x ≤ 4.
Because the function f is not injective, the relation W is multi valued (except for 0). If we limit ourselves to real values of W then the function is only defined for W ≥ −1/e, and has two values in the interval (−1/e, 0). For W ≥ −1 it is the single function W0(x). We have W0(0) = 0 and W0( −1/e) = −1. The lower branch applies to W ≤ −1 and is indicated by W−1(x). It decreases further and further from W−1(−1/e) = −1 to W−1(0−) = −∞.
HistoryThe German-Swiss mathematician Johann Heinrich Lambert (1728 - 1777) described this function. |