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".

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
→
∞
P
(
n
)
F
(
n
)
=
1.
This is symbolically denoted by

P
(
n
)
∼
F
(
n
)
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*):

π
(
x
)
∼
x
log
(
x
)
.
Stirling's approximation is a well-known asymptotic formula for the factorial function:

n
!
=
1
×
2
×
…
×
n
.

The asymptotic formula is

n
!
∼
2
π
n
(
n
e
)
n
.
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. Thus, for example, *P*(4) = 5. G.H. Hardy and Srinivasa Ramanujan in 1918 obtained the following asymptotic formula for *P*(*n*):

P
(
n
)
∼
1
4
n
3
e
π
2
n
/
3
.
The Airy function Ai(x), which is a solution of the differential equation

y
″
−
x
y
=
0
and which has many applications in physics, has the following asymptotic formula:

A
i
(
x
)
∼
e
−
2
3
x
3
/
2
2
π
x
1
/
4
.