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The product logarithm Lambert W function plotted in the complex plane from −2 − 2i to 2 + 2i The graph of y = W(x) for real x < 6 and y > −4.The upper branch (blue) with y ≥ −1 is the graph of the function W 0 (principal branch), the lower branch (magenta) with y ≤ −1 is the graph of the function W −1.
It is the value of W(1), where W is Lambert's W function. The name is derived from the alternate name for Lambert's W function, the omega function. The numerical value of Ω is given by Ω = 0.56714 32904 09783 87299 99686 62210... (sequence A030178 in the OEIS). 1/Ω = 1.76322 28343 51896 71022 52017 76951... (sequence A030797 in the OEIS).
The Lambert W function is the function () that is implicitly defined by the equation () =. We may use the theorem to compute the Taylor series of () at ...
where W represents Lambert's W function. As the limit y = ∞ x (if existent on the positive real line, i.e. for e −e ≤ x ≤ e 1/e) must satisfy x y = y we see that x ↦ y = ∞ x is (the lower branch of) the inverse function of y ↦ x = y 1/y.
An explicit expression for the diode current can be obtained in terms of the Lambert W-function (also called the Omega function). [3] A guide to these manipulations follows. A new variable w {\displaystyle w} is introduced as
In complex analysis, the Omega constant, a solution of Lambert's W function [11] In differential geometry, the space of differential forms on a manifold (of a certain degree, usually with a superscript). A variable for a 2-dimensional region in calculus, usually corresponding to the domain of a double integral.
This is a list of special function eponyms in mathematics, ... Karl L. W. M. Heun (1859 – 1929): ... Johann Heinrich Lambert: Lambert W function;
In mathematics, omega function refers to a function using the Greek letter omega, written ω or Ω. ... , related to the Lambert W Function;