Asymptotic Expression


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Asymptotic Expression

 

a comparatively simple elementary function, approximately equal (with as small a relative error as desired) to a more complex function at large values of the argument (or for argument values that are close to a given value—for example, zero). An asymptotic expression is sometimes called an asymptotic formula or estimate. The exact definition is the following: A function φ (x) is an asymptotic expression for f (x) as x → ∞ (or xa), if f (x) / φ (x) → 1 as x (or xa), or, to put it differently, if f (x) = φ(x) [l + a (x)], where a (x) → 0 as x (or xa). In this case it is written: f (x) ~ φ (x) as xφ (or xa). As a rule, φ (x) must be an easily computable function. The simplest examples of asymptotic expressions for x → 0 are sin xx, tan xx, cot x ∼ 1/x, 1 − cos xx2/2, ln (1 + x) ∼ x, ax − 1 ∼ x In a (a > 0, a ≠ 1). More complex asymptotic expressions as x arise for important functions from the theory of numbers and special functions of mathematical physics. For example, π (x) ∼ x/ln x, where π (x) is the number of prime numbers not exceeding x:

where ┌ (u) is the gamma function; for integer-number values x = n we have ┌(n + 1) = n!, which reduces to Stirling’s formula: Asymptotic Expression as n. Still more complex asymptotic expressions are found, for example in Bessel functions.

Asymptotic expressions are also considered in the complex plane, z = x + iy. Thus, for example, ǀsin(x + iy)ǀ ∼eǀvǀ2 as yφ and yφ.

An asymptotic expression is, in general, a particular case (principal term) of more complex (and accurate) approximate expressions, which are called asymptotic series or expansions.

LITERATURE

Debruijn, N. G.Asimptoticheskie methody v analize. Moscow, 1961. (Translated from English.)

Evgrafov, M. A. Asymptoticheskie otsenki i tselye funktsii, 2nd ed. Moscow, 1962.

V. I. LEVIN

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