Asymptotic formula

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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[edit]

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[edit]

Prime number theorem[edit]

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[edit]

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[edit]

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[edit]

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[edit]

References[edit]

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