Stirling's approximation
E300765
Stirling's approximation is a classical formula in mathematics that provides an efficient asymptotic estimate for factorials and the gamma function, especially for large arguments.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Stirling's approximation canonical | 1 |
| Stirling’s approximation | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2815493 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Stirling's approximation Context triple: [Euler–Maclaurin summation formula, relatedTo, Stirling's approximation]
-
A.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
B.
Hardy–Ramanujan asymptotic formula
The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
-
C.
Pochhammer symbol
The Pochhammer symbol is a mathematical notation representing rising factorials, widely used in series expansions, special functions, and hypergeometric functions.
-
D.
Edgeworth expansion
Edgeworth expansion is an asymptotic series that refines the central limit theorem by providing higher-order approximations to the distribution of normalized sums of random variables.
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E.
Gamma function
The Gamma function is a fundamental extension of the factorial function to complex and real non-integer arguments, widely used in analysis, probability, and mathematical physics.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Stirling's approximation Target entity description: Stirling's approximation is a classical formula in mathematics that provides an efficient asymptotic estimate for factorials and the gamma function, especially for large arguments.
-
A.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
B.
Hardy–Ramanujan asymptotic formula
The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
-
C.
Pochhammer symbol
The Pochhammer symbol is a mathematical notation representing rising factorials, widely used in series expansions, special functions, and hypergeometric functions.
-
D.
Edgeworth expansion
Edgeworth expansion is an asymptotic series that refines the central limit theorem by providing higher-order approximations to the distribution of normalized sums of random variables.
-
E.
Gamma function
The Gamma function is a fundamental extension of the factorial function to complex and real non-integer arguments, widely used in analysis, probability, and mathematical physics.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
asymptotic formula
ⓘ
mathematical approximation ⓘ result in analysis ⓘ |
| appliesTo |
Gamma function
ⓘ
large arguments ⓘ n! ⓘ |
| approximationType |
asymptotic equality
ⓘ
logarithmic approximation ⓘ |
| describes |
asymptotic behavior of factorial function
ⓘ
asymptotic behavior of gamma function ⓘ |
| field |
analytic number theory
ⓘ
asymptotic analysis ⓘ combinatorics ⓘ mathematics ⓘ probability theory ⓘ statistics ⓘ |
| givesApproximation |
ln Γ(z) ≈ (z − 1/2) ln z − z + (1/2) ln(2π)
ⓘ
ln(n!) ≈ n ln n − n ⓘ n! ≈ √(2πn) (n/e)^n ⓘ |
| hasErrorTerm | relative error tends to 0 as n → ∞ ⓘ |
| hasExponentialFactor | (n/e)^n ⓘ |
| hasFormulation |
logarithmic form
ⓘ
multiplicative form ⓘ series expansion form ⓘ |
| hasHigherOrderTerm |
1/(1260n^5)
ⓘ
1/(12n) ⓘ −1/(360n^3) ⓘ |
| hasLeadingFactor | √(2πn) ⓘ |
| hasRefinement |
Euler–Maclaurin summation formula
ⓘ
surface form:
Stirling series
|
| historicalPeriod | 18th century ⓘ |
| improvesOn | earlier approximations of factorials ⓘ |
| isAsymptoticExpansionOf | ln Γ(z) ⓘ |
| namedAfter | James Stirling ⓘ |
| relatedTo |
Euler’s reflection formula
ⓘ
surface form:
Gamma function reflection formula
Laplace method ⓘ
surface form:
Laplace's method
central limit theorem ⓘ entropy of multinomial coefficients ⓘ saddle-point method ⓘ |
| usedFor |
approximating binomial coefficients
ⓘ
approximating log-factorials in statistics ⓘ asymptotic estimates in combinatorics ⓘ complexity estimates in algorithms ⓘ deriving normal approximations to distributions ⓘ entropy calculations ⓘ estimating factorials ⓘ |
| usedIn |
approximation of partition functions in statistical mechanics
ⓘ
derivation of entropy formula in information theory ⓘ |
| validFor |
complex z with |arg z| < π
ⓘ
positive integers n ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Stirling's approximation Description of subject: Stirling's approximation is a classical formula in mathematics that provides an efficient asymptotic estimate for factorials and the gamma function, especially for large arguments.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Stirling’s approximation