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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:
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which is often written as
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(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.
Speed of convergence and error estimates
More precisely,
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with
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Stirling's formula is in fact the first approximation to the following series (now called the Stirling
series):
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As , the error in the truncated series
is asymptotically equal to the first omitted term. This is an example of an asymptotic expansion.
Derivation
The formula, together with precise estimates of its error, can be derived as follows. Instead of approximating n!,
one considers the natural logarithm ln(n!) = ln(1) +
ln(2) + ... + ln(n); the Euler-Maclaurin
formula gives estimates for sums like these. The goal, then, is to show the approximation formula in its logarithmic
form:
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Alternatively, the leading term of Stirling's approximation can be obtained through the method of steepest descent.
History
The formula was first discovered by Abraham de Moivre in the
form
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Stirling's contribution consisted of showing that the "constant" is .
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