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The omega constant is a mathematical constant defined as the unique real number that satisfies the equation = 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
For example, the constant π may be defined as the ratio of the length of a circle's circumference to its diameter. The following list includes a decimal expansion and set containing each number, ordered by year of discovery. The column headings may be clicked to sort the table alphabetically, by decimal value, or by set.
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.
The function F is called universal if for every computable function f of a single variable there is a string w such that for all x, F(w x) = f(x); here w x represents the concatenation of the two strings w and x. This means that F can be used to simulate any computable function of one
the arithmetic function counting a number's distinct prime factors [64] the symbol ϖ, a graphic variant of π, is sometimes construed as omega with a bar over it; see π; the unsaturated fats nomenclature in biochemistry (e.g. ω−3 fatty acids) the first uncountable ordinal (also written as Ω)
The ordinary generating function of a sequence can be expressed as a rational function (the ratio of two finite-degree polynomials) if and only if the sequence is a linear recursive sequence with constant coefficients; this generalizes the examples above. Conversely, every sequence generated by a fraction of polynomials satisfies a linear ...
where α and β are real sequences which decay fast enough to provide the convergence of the series, at least at moderate values of Im z. The function S satisfies the tetration equations S(z + 1) = exp(S(z)), S(0) = 1, and if α n and β n approach 0 fast enough it will
Oscillation of a sequence (shown in blue) is the difference between the limit superior and limit inferior of the sequence. In mathematics, the oscillation of a function or a sequence is a number that quantifies how much that sequence or function varies between its extreme values as it approaches infinity or a point.