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

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

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

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

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

References

1. "Sci-Tech Dictionary: asymptotic formula". Retrieved 13 May 2010.
2. Weisstein, Eric W. "Partition Function P." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PartitionFunctionP.html