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 x → a), if f (x) / φ (x) → 1 as x → ∞ (or x → a), or, to put it differently, if f (x) = φ(x) [l + a (x)], where a (x) → 0 as x → ∞ (or x → a). In this case it is written: f (x) ~ φ (x) as x → φ (or x → a). As a rule, φ (x) must be an easily computable function. The simplest examples of asymptotic expressions for x → 0 are sin x ∼ x, tan x ∼ x, cot x ∼ 1/x, 1 − cos x ∼ x²/2, ln (1 + x) ∼ x, a^x − 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 ∼eǀ^vǀ2 as y → φ and y → φ.

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.