Lambert W function

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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) = wew where ew 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 real-valued W then the relation is defined only for x ≥ −1/e, and is double-valued on (−1/e, 0); the additional constraint W ≥ −1 defines a single-valued function W0(x). We have W0(0) = 0 and W0(−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).

Contents

Terminology

The Lambert W-function is named after Johann Heinrich Lambert. The main branch W0 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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