Hardy notation

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In complexity theory and mathematics, the Hardy notation, introduced by G. H. Hardy, is used for asymptotic comparison of functions, equivalently to Landau notation (also known as "Big O notation").

It is defined in terms of Landau notation by

$f\lesssim g \iff f \in O(g)$ and $f\ll g \iff f\in o(g).$

(Similar symbols are used, like $\preceq$ resp. $\prec\!\!\!\!\!\!\!\!\prec$ .)

The Hardy notation is commonly abused similarly to the Landau notation. For example, where $h 2 = O (h 3)$ is the shortened/abused Landau notation of $h\mapsto h^2 \in O(h\mapsto h^3)$ , the expression $h^2 \lesssim h^3$ is the shortened/abused Hardy notation of $h\mapsto h^2 \lesssim h\mapsto h^3$ .

For more examples and applications, see Landau notation and references therein.

[edit] See also

Big O notation: more explicit definitions, explanations, properties and related notations for real valued functions
Taylor's theorem, maybe the most important application of Landau notation in mathematical analysis
Asymptotic expansion: approximation of functions generalizing Taylor's formula.
Nachbin's theorem: a precise way of bounding complex analytic functions so that the domain of convergence of integral transforms can be stated.

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Categories: Mathematical notation | Asymptotic analysis

Hardy notation

From Wikipedia, the free encyclopedia

[edit] See also

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