# 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 *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* ∼ *x*^{2}/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^{v}ǀ2 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.

### 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