Laplace method
E157382
The Laplace method is an asymptotic technique in mathematical analysis used to approximate integrals, especially those dominated by contributions near a maximum point of the integrand.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Laplace approximation | 4 |
| Laplace method canonical | 2 |
| Laplace's method | 1 |
| Laplace’s approximation | 1 |
| Watson’s lemma | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1382341 — 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: Laplace method Context triple: [Gaussian integral, relatedTo, Laplace method]
-
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.
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B.
Darwin–Fowler method
The Darwin–Fowler method is a statistical mechanics technique that uses complex analysis and generating functions to derive distribution laws for systems of many particles.
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C.
Laplace transform
The Laplace transform is an integral transform widely used in mathematics, physics, and engineering to convert functions of time into functions of a complex variable, simplifying the analysis and solution of differential equations and linear systems.
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D.
Euler’s method for numerical integration
Euler’s method for numerical integration is a simple first-order numerical procedure used to approximate solutions to ordinary differential equations by stepping forward in small increments.
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E.
Laplace operator
The Laplace operator is a second-order differential operator widely used in mathematics and physics to describe phenomena such as diffusion, heat flow, and wave propagation.
- 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: Laplace method Target entity description: The Laplace method is an asymptotic technique in mathematical analysis used to approximate integrals, especially those dominated by contributions near a maximum point of the integrand.
-
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.
Darwin–Fowler method
The Darwin–Fowler method is a statistical mechanics technique that uses complex analysis and generating functions to derive distribution laws for systems of many particles.
-
C.
Laplace transform
The Laplace transform is an integral transform widely used in mathematics, physics, and engineering to convert functions of time into functions of a complex variable, simplifying the analysis and solution of differential equations and linear systems.
-
D.
Euler’s method for numerical integration
Euler’s method for numerical integration is a simple first-order numerical procedure used to approximate solutions to ordinary differential equations by stepping forward in small increments.
-
E.
Laplace operator
The Laplace operator is a second-order differential operator widely used in mathematics and physics to describe phenomena such as diffusion, heat flow, and wave propagation.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
asymptotic method
ⓘ
method for asymptotic evaluation of integrals ⓘ technique in mathematical analysis ⓘ |
| appliesTo |
integrals with a large parameter in the exponent
ⓘ
real-valued integrals ⓘ |
| approximationFormula | ∫_a^b e^{λ f(x)} g(x) dx ≈ e^{λ f(x0)} g(x0) √(2π/(−λ f''(x0))) for large λ ⓘ |
| assumes |
existence of a point where the integrand attains a maximum
ⓘ
nondegenerate maximum of the phase function ⓘ smoothness of the integrand near the maximum point ⓘ |
| basedOn |
Gaussian approximation near the maximum
ⓘ
Taylor expansion of the exponent around a maximum point ⓘ |
| category |
asymptotic expansion technique
ⓘ
integral approximation method ⓘ |
| characterizedBy |
dominant contribution from neighborhoods of maxima
ⓘ
exponential accuracy in the large-parameter limit ⓘ |
| contrastedWith |
saddle point method in the complex plane
ⓘ
stationary phase method for oscillatory integrals ⓘ |
| field |
applied mathematics
ⓘ
asymptotic analysis ⓘ mathematical analysis ⓘ |
| historicalPeriod | 19th century ⓘ |
| influenced |
development of asymptotic expansion techniques
ⓘ
modern methods of steepest descent ⓘ |
| namedAfter | Pierre-Simon Laplace ⓘ |
| relatedTo |
Laplace’s approximation
ⓘ
linked to:
Laplace method
method of steepest descent ⓘ saddle point method ⓘ stationary phase method ⓘ |
| requires |
identification of the global maximum of the phase function on the integration domain
ⓘ
second derivative of the phase function at the maximum to be negative ⓘ |
| typicalForm | ∫_a^b e^{λ f(x)} g(x) dx with λ → +∞ ⓘ |
| usedFor |
approximating integrals
ⓘ
approximating integrals dominated by a maximum of the integrand ⓘ approximating integrals dominated by a stationary point ⓘ approximating integrals of the form ∫ e^{λ f(x)} g(x) dx as λ → ∞ ⓘ asymptotic expansion of integrals ⓘ evaluating integrals with large parameters ⓘ |
| usedIn |
Bayesian statistics
ⓘ
linked to:
Bayesian inference
approximation of likelihood integrals ⓘ approximation of partition functions ⓘ large deviations theory ⓘ probability theory ⓘ statistical mechanics ⓘ statistics ⓘ |
| x0 | point where f(x) attains its maximum on [a,b] ⓘ |
| yields |
higher-order asymptotic corrections when extended
ⓘ
leading-order asymptotic term of an integral ⓘ |
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: Laplace method Description of subject: The Laplace method is an asymptotic technique in mathematical analysis used to approximate integrals, especially those dominated by contributions near a maximum point of the integrand.
Referenced by (9)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Laplace method
linked to: Laplace method
linked to: Laplace method
linked to: Laplace method
linked to: Laplace method
linked to: Laplace method
linked to: Laplace method