Gaussian integral
E29365
The Gaussian integral is a fundamental result in mathematics that evaluates the integral of the exponential of a negative quadratic function over the entire real line, yielding a value proportional to the square root of π and underpinning the normal distribution in probability theory.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Gaussian integral canonical | 1 |
| Gaussian integrals | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T228925 — 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: Gaussian integral Context triple: [Carl Friedrich Gauss, notableWork, Gaussian integral]
-
A.
Feynman path integral
The Feynman path integral is a formulation of quantum mechanics in which a particle’s behavior is described as a sum over all possible paths it can take, each weighted by a phase factor derived from the classical action.
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B.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
-
C.
Born–Huang expansion
The Born–Huang expansion is a quantum mechanical method that systematically improves upon the Born–Oppenheimer approximation by including couplings between electronic and nuclear motions in molecular systems.
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D.
Fokker–Planck equation
The Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of a stochastic (random) process, such as Brownian motion.
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E.
Feynman–Hellmann theorem
The Feynman–Hellmann theorem is a result in quantum mechanics that relates the derivative of an energy eigenvalue with respect to a parameter in the Hamiltonian to the expectation value of the corresponding derivative of the Hamiltonian.
- 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: Gaussian integral Target entity description: The Gaussian integral is a fundamental result in mathematics that evaluates the integral of the exponential of a negative quadratic function over the entire real line, yielding a value proportional to the square root of π and underpinning the normal distribution in probability theory.
-
A.
Feynman path integral
The Feynman path integral is a formulation of quantum mechanics in which a particle’s behavior is described as a sum over all possible paths it can take, each weighted by a phase factor derived from the classical action.
-
B.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
-
C.
Born–Huang expansion
The Born–Huang expansion is a quantum mechanical method that systematically improves upon the Born–Oppenheimer approximation by including couplings between electronic and nuclear motions in molecular systems.
-
D.
Fokker–Planck equation
The Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of a stochastic (random) process, such as Brownian motion.
-
E.
Feynman–Hellmann theorem
The Feynman–Hellmann theorem is a result in quantum mechanics that relates the derivative of an energy eigenvalue with respect to a parameter in the Hamiltonian to the expectation value of the corresponding derivative of the Hamiltonian.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
definite integral
ⓘ
mathematical concept ⓘ |
| appearsIn |
Brownian motion theory
ⓘ
central limit theorem proofs ⓘ heat equation solutions ⓘ |
| category |
closed-form integrals
ⓘ
special integrals ⓘ |
| condition | a > 0 in ∫_{−∞}^{∞} e^{−a x^2} dx ⓘ |
| consequence | moments of the normal distribution are finite ⓘ |
| convergenceReason | rapid decay of e^{−x^2} at infinity ⓘ |
| definedAs | ∫_{−∞}^{∞} e^{−x^2} dx ⓘ |
| dimension | one-dimensional case of Gaussian measures ⓘ |
| domainOfIntegration | (−∞, ∞) ⓘ |
| evaluationMethod |
squaring the integral and using polar coordinates
ⓘ
using the Gamma function Γ(1/2) = √π ⓘ |
| field |
mathematical analysis
ⓘ
probability theory ⓘ statistics ⓘ |
| generalization | ∫_{−∞}^{∞} e^{−a x^2} dx = √(π/a) ⓘ |
| historicalAttribution | Carl Friedrich Gauss ⓘ |
| implies | ∫_{0}^{∞} e^{−x^2} dx = √π / 2 ⓘ |
| integrand | e^{−x^2} ⓘ |
| notation | ∫_{−∞}^{∞} e^{−x^2} dx = √π ⓘ |
| property | convergent improper integral ⓘ |
| relatedTo |
Gamma function
ⓘ
Laplace method ⓘ error function ⓘ multidimensional Gaussian integral ⓘ normal distribution ⓘ saddle-point approximation ⓘ standard normal distribution ⓘ |
| requires | a > 0 for convergence in ∫_{−∞}^{∞} e^{−a x^2} dx ⓘ |
| role |
basis for defining Gaussian measure
ⓘ
normalization constant for Gaussian distributions ⓘ |
| symmetryProperty | even integrand ⓘ |
| type |
Lebesgue integration
ⓘ
surface form:
Lebesgue integral
improper Riemann integral ⓘ |
| usedFor |
approximating sums by integrals in asymptotic analysis
ⓘ
computing partition functions of quadratic Hamiltonians ⓘ evaluating Fresnel-type integrals via transformations ⓘ |
| usedIn |
Fourier analysis
ⓘ
derivation of the normal distribution ⓘ error function definition ⓘ path integrals ⓘ probability density normalization ⓘ quantum mechanics ⓘ statistical mechanics ⓘ |
| value | √π ⓘ |
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: Gaussian integral Description of subject: The Gaussian integral is a fundamental result in mathematics that evaluates the integral of the exponential of a negative quadratic function over the entire real line, yielding a value proportional to the square root of π and underpinning the normal distribution in probability theory.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Gaussian integrals