Math 2.0

Approximations and expansions Definition

Asymptotic equivalence

Builds on Little o notation

Definition

Let ff and gg be real functions defined on an interval [x0,∞)[x_0, \infty). The functions are asymptotically equivalent as x→∞x \to \infty if f(x)−g(x)=o(g(x))f(x) - g(x) = o(g(x)) as x→∞x \to \infty, in the sense of little o notation. The notation is f(x)∼g(x)f(x) \sim g(x).

Remarks

If g(x)≠0g(x) \ne 0 for large xx, then f∼gf \sim g means that f(x)/g(x)→1f(x)/g(x) \to 1. The relation ∼\sim is an equivalence relation. For example, x2+x∼x2x^2 + x \sim x^2. Stirling's formula states n!∼2πn (n/e)nn! \sim \sqrt{2 \pi n}\, (n/e)^n. Mathlib defines it for filters, as Asymptotics.IsEquivalent.

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  1. Big O notation definition
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Where this leads

8 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Asymptotic_analysis, Definition
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