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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
The notation convention chosen here (with W 0 and W −1) follows the canonical reference on the Lambert W function by Corless, Gonnet, Hare, Jeffrey and Knuth. [3]The name "product logarithm" can be understood as follows: since the inverse function of f(w) = e w is termed the logarithm, it makes sense to call the inverse "function" of the product we w the "product logarithm".
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]
the omega constant 0.5671432904097838729999686622... an asymptotic lower bound notation related to big O notation; in probability theory and statistical mechanics, the support; a solid angle; the omega baryon; the arithmetic function counting a number's prime factors counted with multiplicity; the density parameter in cosmology
Let k = 1/Ω, that is, the reciprocal of the Omega constant. Then it holds that: = That is, it imitates the power rule of derivatives (even though it's not a polynomial, but an exponential function with base k) in reducing the exponent by 1. The reciprocal Omega constant is the unique exponential base for which this is true.
Any non-computable number, in particular: Chaitin's constant. Constructed irrational numbers which are not simply normal in any base. [30] Any number for which the digits with respect to some fixed base form a Sturmian word. [31] The Prouhet–Thue–Morse constant [32] and the related rabbit constant. [33] The Komornik–Loreti constant. [34]
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The circumference of a circle with diameter 1 is π.. A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]