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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.