
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Stirling's approximation (or Stirling's formula) is an
asymptotic
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates Limit of a function#Limits at infinity, tends to infinity. In pro ...
approximation for
factorial
In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times ...
s. It is a good approximation, leading to accurate results even for small values of
. It is named after
James Stirling, though a related but less precise result was first stated by
Abraham de Moivre.
One way of stating the approximation involves the
logarithm
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
of the factorial:
where the
big O notation
Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a memb ...
means that, for all sufficiently large values of
, the difference between
and
will be at most proportional to the logarithm of
. In computer science applications such as the
worst-case lower bound for comparison sorting, it is convenient to instead use the
binary logarithm
In mathematics, the binary logarithm () is the exponentiation, power to which the number must be exponentiation, raised to obtain the value . That is, for any real number ,
:x=\log_2 n \quad\Longleftrightarrow\quad 2^x=n.
For example, th ...
, giving the equivalent form
The error term in either base can be expressed more precisely as
, corresponding to an approximate formula for the factorial itself,
Here the sign
means that the two quantities are asymptotic, that is, their ratio tends to 1 as
tends to infinity.
Derivation
Roughly speaking, the simplest version of Stirling's formula can be quickly obtained by approximating the sum
with an
integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
:
The full formula, together with precise estimates of its error, can be derived as follows. Instead of approximating
, one considers its
natural logarithm
The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
, as this is a
slowly varying function:
The right-hand side of this equation minus
is the approximation by the
trapezoid rule of the integral
and the error in this approximation is given by the
Euler–Maclaurin formula:
where
is a
Bernoulli number, and is the remainder term in the Euler–Maclaurin formula. Take limits to find that
Denote this limit as
. Because the remainder in the Euler–Maclaurin formula satisfies
where
big-O notation is used, combining the equations above yields the approximation formula in its logarithmic form:
Taking the exponential of both sides and choosing any positive integer
, one obtains a formula involving an unknown quantity
. For , the formula is
The quantity
can be found by taking the limit on both sides as
tends to infinity and using
Wallis' product, which shows that
. Therefore, one obtains Stirling's formula:
Alternative derivations
An alternative formula for
using the
gamma function
In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
is
(as can be seen by repeated integration by parts). Rewriting and changing variables , one obtains
Applying
Laplace's method one has
which recovers Stirling's formula:
Higher orders
In fact, further corrections can also be obtained using Laplace's method. From previous result, we know that
, so we "peel off" this dominant term, then perform two changes of variables, to obtain:
To verify this:
.
Now the function
is unimodal, with maximum value zero. Locally around zero, it looks like
, which is why we are able to perform Laplace's method. In order to extend Laplace's method to higher orders, we perform another change of variables by
. This equation cannot be solved in closed form, but it can be solved by serial expansion, which gives us
. Now plug back to the equation to obtain
notice how we don't need to actually find
, since it is cancelled out by the integral. Higher orders can be achieved by computing more terms in
, which can be obtained programmatically.
Thus we get Stirling's formula to two orders:
Complex-analytic version
A complex-analysis version of this method is to consider
as a
Taylor coefficient of the exponential function
, computed by
Cauchy's integral formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
as
This line integral can then be approximated using the
saddle-point method with an appropriate choice of contour radius
. The dominant portion of the integral near the saddle point is then approximated by a real integral and Laplace's method, while the remaining portion of the integral can be bounded above to give an error term.
Using the Central Limit Theorem and the Poisson distribution
An alternative version uses the fact that the
Poisson distribution
In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...
converges to a
normal distribution
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
f(x) = \frac ...
by the
Central Limit Theorem.
Since the Poisson distribution with parameter
converges to a normal distribution with mean
and variance
, their
density functions will be approximately the same:
Evaluating this expression at the mean, at which the approximation is particularly accurate, simplifies this expression to:
Taking logs then results in:
which can easily be rearranged to give:
Evaluating at
gives the usual, more precise form of Stirling's approximation.
Speed of convergence and error estimates

Stirling's formula is in fact the first approximation to the following series (now called the Stirling series):
An explicit formula for the coefficients in this series was given by G. Nemes. Further terms are listed in the
On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to th ...
as and . The first graph in this section shows the
relative error vs.
, for 1 through all 5 terms listed above. (Bender and Orszag p. 218) gives the asymptotic formula for the coefficients:
which shows that it grows superexponentially, and that by the
ratio test
In mathematics, the ratio test is a convergence tests, test (or "criterion") for the convergent series, convergence of a series (mathematics), series
:\sum_^\infty a_n,
where each term is a real number, real or complex number and is nonzero wh ...
the
radius of convergence is zero.

As , the error in the truncated series is asymptotically equal to the first omitted term. This is an example of an
asymptotic expansion
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation ...
. It is not a
convergent series; for any ''particular'' value of
there are only so many terms of the series that improve accuracy, after which accuracy worsens. This is shown in the next graph, which shows the relative error versus the number of terms in the series, for larger numbers of terms. More precisely, let be the Stirling series to
terms evaluated at
. The graphs show
which, when small, is essentially the relative error.
Writing Stirling's series in the form
it is known that the error in truncating the series is always of the opposite sign and at most the same magnitude as the first omitted term.
Other bounds, due to Robbins, valid for all positive integers
are
This upper bound corresponds to stopping the above series for
after the
term. The lower bound is weaker than that obtained by stopping the series after the
term. A looser version of this bound is that
for all
.
Stirling's formula for the gamma function
For all positive integers,
where denotes the
gamma function
In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
.
However, the gamma function, unlike the factorial, is more broadly defined for all complex numbers other than non-positive integers; nevertheless, Stirling's formula may still be applied. If , then
Repeated integration by parts gives
where
is the
th
Bernoulli number (note that the limit of the sum as
is not convergent, so this formula is just an
asymptotic expansion
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation ...
). The formula is valid for
large enough in absolute value, when , where is positive, with an error term of . The corresponding approximation may now be written:
where the expansion is identical to that of Stirling's series above for
, except that
is replaced with .
A further application of this asymptotic expansion is for complex argument with constant . See for example the Stirling formula applied in of the
Riemann–Siegel theta function on the straight line .
A convergent version of Stirling's formula
Thomas Bayes
Thomas Bayes ( , ; 7 April 1761) was an English statistician, philosopher and Presbyterian minister who is known for formulating a specific case of the theorem that bears his name: Bayes' theorem.
Bayes never published what would become his m ...
showed, in a letter to
John Canton published by the
Royal Society
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
in 1763, that Stirling's formula did not give a
convergent series. Obtaining a convergent version of Stirling's formula entails evaluating
Binet's formula:
One way to do this is by means of a convergent series of inverted
rising factorials. If
then
where
where denotes the
Stirling numbers of the first kind. From this one obtains a version of Stirling's series
which converges when .
Stirling's formula may also be given in convergent form as
where
Versions suitable for calculators
The approximation
and its equivalent form
can be obtained by rearranging Stirling's extended formula and observing a coincidence between the resultant
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
and the
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
expansion of the
hyperbolic sine
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a unit circle, circle with a unit radius, the points form the right ha ...
function. This approximation is good to more than 8 decimal digits for with a real part greater than 8. Robert H. Windschitl suggested it in 2002 for computing the gamma function with fair accuracy on calculators with limited program or register memory.
Gergő Nemes proposed in 2007 an approximation which gives the same number of exact digits as the Windschitl approximation but is much simpler:
or equivalently,
An alternative approximation for the gamma function stated by
Srinivasa Ramanujan in
Ramanujan's lost notebook is
for . The equivalent approximation for has an asymptotic error of and is given by
The approximation may be made precise by giving paired upper and lower bounds; one such inequality is
History
The formula was first discovered by
Abraham de Moivre in the form
De Moivre gave an approximate rational-number expression for the natural logarithm of the constant. Stirling's contribution consisted of showing that the constant is precisely
.
See also
*
Lanczos approximation
*
Spouge's approximation
References
Further reading
*
*
*
*
*
External links
*
Peter Luschny, ''Approximation formulas for the factorial function n!''*
*
{{Calculus topics
Approximations
Asymptotic analysis
Analytic number theory
Gamma and related functions
Theorems in mathematical analysis