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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 standard Lambert W function expresses exact solutions to what is called ``transcendental algebraic equations of the form:. exp(-c*x) = a_o*(x-r) where a_o, c, and r are real constants.
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.
The Lambert W function has several examples, but only has proof for the first one. Does anyone have a proof for example 3? —Preceding unsigned comment added by Luckytoilet (talk • contribs) 05:05, 17 February 2010 (UTC) By continuity of exponentiation, the limit c satisfies c = z c = e c log z.
Lambert (grape), another name for the German/Italian wine grape Trollinger; Lambert v. California, court case regarding legal notice; Lambert W function, mathematical definition of a product log named after Johann Heinrich Lambert; Lambert the Sheepish Lion, a 1952 Disney animated short film directed by Jack Hannah
The range of the Lambert W function, showing all branches. The orange curves are images of either the positive or the negative imaginary axis. The black curves are images of the positive or negative real axis (except for the one that intersects −1, which is the image of part of the negative real axis).
Lambert (lunar crater). [1] In the MARE IMBRIUM, Diameter: 30.1209 km; Lambert (Martian crater). [1] In the Sinus Sabaeus quadrangle of Mars, located at 20.2°S latitude and 334.7°W longitude. It is 92.0 km in diameter