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]

1 Definition
2 Examples
3 See also
4 References

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 toF(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

Stirling's approximation formula is a well known asymptotic formula for the following quantity:

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

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