Asymptotic Expression

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: 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

References in periodicals archive ?
1], that is we must maximize the asymptotic expression
For the general renewal risk model, an asymptotic expression For the finite time ruin probability is presented in [6]; that is,
The reason of the enhancement at 120[degrees] is the result of the asymptotic expression which is given in Equation (51).
With these parameter values, and using the asymptotic expression for the t-statistic (not shown), it is easy to calculate that, as [EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII].
Similarly from Equation (46) we have the next asymptotic expression for the electric field
Qrtega in [2] have studied the dynamics of a class of mappings defined on the plane, which have an asymptotic expression
n]) can be used for all n [is greater than] 3 as long as t [is less than] 1 (t = h/n), whereas the asymptotic expression (37) (denoted by [Y.
Inserting the asymptotic expression for [Mathematical Expression Omitted] (eq.
where B(a, b) = [GAMMA](a) [GAMMA] (b)/ [GAMMA] (a + b) and [PSI](x)is the digamma function, the asymptotic expression of SOP at high MER can be written as
Suppose that the moment generating function satisfies the asymptotic expression
Concerning the convergence of the infinite series in (5), substitute an asymptotic expression for [K.

Site: Follow: Share:
Open / Close