In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for large factorials. It is named in honour of James Stirling. Formally, it states: History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... In mathematics, the factorial of a natural number n is the product of the positive integers less than or equal to n. ... James Stirling (April 22, 1692–December 5, 1770) was an important Scottish mathematician. ...

which is often written as

(See limit, square root, π, e.) For large n, the right hand side is a good approximation for n!, and much faster and easier to calculate. For example, the formula gives for 30! the approximation 2.6452 × 10³² while the correct value is about 2.6525 × 10³². The error is less than 0.3% in this case. In mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets close to either some point, or infinity; or the behavior of a sequences elements, as their index approaches infinity. ... In mathematics, the principal square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is . ... The minuscule, or lower-case, pi The mathematical constant π represents the ratio of a circles circumference to its diameter and is commonly used in mathematics, physics, and engineering. ... The mathematical constant e (occasionally called Eulers number after the Swiss mathematician Leonhard Euler, or Napiers constant in honor of the Scottish mathematician John Napier who introduced logarithms) is the base of the natural logarithm function. ...

Derivation

The formula, together with precise estimates of its error, can be derived as follows. Instead of approximating n!, one considers the natural logarithm The natural logarithm is the logarithm to the base e, where e is approximately equal to 2. ...

Then, we can apply Euler-Maclaurin formula by putting f(x) = ln(x) to find a approximation of the value of ln(n!). In mathematics, the Euler-Maclaurin formula provides a powerful connection between integrals (see calculus) and sums. ...

where B_k is Bernoulli number and R is the remainder of Euler-Maclaurin formula. In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ...

Then, we can then take limit on both sides,

Let the above limit be y and compound the above two formula, we get the approximation formula in its logarithmic form:

where O(f(n)) is Big-O notation. In complexity theory, computer science, and mathematics the Big O notation is a mathematical notation used to describe the asymptotic behavior of functions. ...

Just take exponential on both sides, and take an positive integer m, say 1. We got the formula with an unknown term e^y.

The unknown term e^y can be found by taking limit on both side as n tends to infinity and using Wallis' product. One can estimate the value of e^y is . Therefore, we got Stirling's formula: In mathematics, Wallis product for π states that Proof First of all, consider the root of sin(x)/x is ±nπ, where n = 1, 2, 3, ... Then, we can express sine as an infinite product of linear factors given by its roots: To find the constant k, taking limit on...

The formula may also be obtained by repeated integration by parts. The leading term can be found through the method of steepest descent. In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ... For the optimization method called steepest descent see gradient descent. ...

Speed of convergence and error estimates

More precisely,

with

Stirling's formula is in fact the first approximation to the following series (now called the Stirling series):

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 and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ...

The asymptotic expansion of the logarithm is also called Stirling's series:

In this case, it is known that the error in truncating the series is always of the same sign and at most the same magnitude as the first omitted term.

Stirling's formula for the Gamma function

Stirling's formula may also be applied to the Gamma function The Gamma function along an interval In mathematics, the Gamma function is a function that extends the concept of factorial to the complex numbers. ...

Γ(z + 1) = Π(z) = z!

defined for all complex numbers other than non-positive integers. If then

Repeated integration by parts gives the asymptotic expansion

where B_n is the nth Bernoulli number. The formula is valid for z large enough in absolute value when , where ε is positive, with an error term of O(z ^{− m − 1 / 2}) when the first m terms are used. In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ...

A convergent version of Stirling's formula

Obtaining a convergent version of Stirling's formula entails evaluating

One way to do this is by means of a convergent series of inverted rising exponentials. If , then In mathematics, the Pochhammer symbol, introduced by Leo August Pochhammer, is used in the theory of special functions to represent the rising factorial or upperfactorial and, confusingly, is used in combinatorics to represent the falling factorial or lower factorial . The empty product (x)0 is defined to be 1 in...

where

From this we obtain a version of Stirling's series

which converges when .

History

The formula was first discovered by Abraham de Moivre in the form Abraham de Moivre (May 26, 1667 - November 27, 1754), was a French mathematician famous for de Moivres formula, which links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. ...

Stirling's contribution consisted of showing that the "constant" is . The more precise versions are due to Jacques Binet.

References

• Abromowitz, M. and Stegun, I., Handbook of Mathematical Functions, http://www.math.hkbu.edu.hk/support/aands/toc.htm
• Paris, R. B., and Kaminsky, D., Asymptotics and the Mellin-Barnes Integrals, Cambridge University Press, 2001
• Whittaker, E. T., and Watson, G. N., A Course in Modern Analysis, fourth edition, Cambridge University Press, 1963. ISBN 0521588073