Knowra Stirling's approximation Stirling's approximation Stirling's approximation estimates factorials for large integers: n! ∼ √(2πn)(n/e)ⁿ. Its leading terms also describe the asymptotic growth of the gamma function.
Gamma function : A function that extends factorials to complex arguments, satisfying Γ(n+1)=n! for nonnegative integers n. Its asymptotic behavior is the continuous counterpart of the factorial formula.
Binomial distribution : The probability distribution for the number of successes in a fixed number of independent trials with equal success probability. Approximating its factorials gives useful estimates for probabilities and tail behavior.
Factorial : For a nonnegative integer n, the product of all positive integers up to n, with 0! defined as 1. Stirling's approximation estimates this rapidly growing sequence.
Leading-order asymptotics : An approximation retaining only the dominant term or factors in a quantity's limiting behavior. The familiar factorial formula keeps the leading scale and omits smaller corrections.
Euler–Maclaurin formula : A summation formula that relates sums to integrals and adds correction terms involving derivatives. Applying it to the logarithm of a factorial yields Stirling's expansion.
Entropy : A measure of uncertainty or multiplicity, defined differently across information theory, statistical mechanics, and mathematics. Taking logarithms of factorial counts turns combinatorial growth into entropy-like expressions.
Asymptotic analysis : The study of how functions behave as an input approaches a limit, especially infinity. The symbol ∼ states that the ratio of the two sides tends to one.
Log-gamma function : The natural logarithm of the gamma function, often computed directly to avoid overflow. Numerical work often uses log Γ(n+1) instead of approximating or constructing n!.
Laplace's method : An asymptotic technique for estimating integrals dominated by contributions near a maximum or minimum. It derives the leading factorial estimate from an integral representation of the gamma function.
Central binomial coefficient : The coefficient binom(2n,n), which counts ways to choose n objects from 2n. Stirling's approximation shows it grows like 4ⁿ/√(πn).
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