Asymptotic formula

In mathematics, an asymptotic formula for a quantity (function or expression) depending on natural numbers, or on a variable taking real numbers as values, is a function of natural numbers, or of a real variable, whose values are nearly equal to the values of the former when both are evaluated for the same large values of the variable. An asymptotic formula for a quantity is a function which is asymptotically equivalent to the former.

More generally, an asymptotic formula is "a statement of equality between two functions which is not a true equality but which means the ratio of the two functions approaches 1 as the variable approaches some value, usually infinity".

Definition

Let P(n) be a quantity or function depending on n which is a natural number. A function F(n) of n is an asymptotic formula for P(n) if P(n) is asymptotically equivalent to F(n), that is, if

$\lim_{n\rightarrow \infty}\frac{P(n)}{F(n)}=1.$

This is symbolically denoted by

$P(n) \sim F(n)\,$

Examples

Prime number theorem

For a real number x, let π (x) denote the number of prime numbers less than or equal to x. The classical prime number theorem gives an asymptotic formula for π (x):

$\pi(x)\sim \frac{x}{\log(x)}.$

Stirling's formula

File:Stirling's Approximation.svg

Stirling's approximation approaches the factorial function as n increases.

Stirling's approximation is a well-known asymptotic formula for the factorial function:

$n!=1\times 2\times\ldots \times n$.

The asymptotic formula is

$n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n.$

Asymptotic formula for the partition function

For a positive integer n, the partition function P(n), sometimes also denoted p(n), gives the number of ways of writing the integer n as a sum of positive integers, where the order of addends is not considered significant. Thus, for example, P(4) = 5. G.H. Hardy and Srinivasa Ramanujan in 1918 obtained the following asymptotic formula for P(n):

$P(n)\sim \frac{1}{4n\sqrt{3}} e^{\pi\sqrt{2n/3}}.$

Asymptotic formula for Airy function

The Airy function Ai(x), which is a solution of the differential equation

$y''-xy=0\,$

and which has many applications in physics, has the following asymptotic formula:

$\mathrm{Ai}(x) \sim \frac{e^{-\frac{2}{3}x^{3/2}}}{2\sqrt{\pi}x^{1/4}}.$

See also

• Asymptotic analysis