# 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".[1]

## 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

${\displaystyle \lim _{n\rightarrow \infty }{\frac {P(n)}{F(n)}}=1.}$

This is symbolically denoted by

${\displaystyle P(n)\sim F(n)\,}$

## 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):

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

### Stirling's formula

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

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

${\displaystyle n!=1\times 2\times \ldots \times n}$.

The asymptotic formula is

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

### 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.[2] Thus, for example, P(4) = 5. G.H. Hardy and Srinivasa Ramanujan in 1918 obtained the following asymptotic formula for P(n):[2]

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

### Asymptotic formula for Airy function

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

${\displaystyle y''-xy=0\,}$

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

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