# 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

This is symbolically denoted by

## 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*):

### Stirling's formula

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

- .

The asymptotic formula is

### 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]}

### Asymptotic formula for Airy function

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

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

## See also

## References

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