Fresnel integrals
E198136
Fresnel integrals are special functions in mathematics that describe the complex oscillatory behavior of wave diffraction and interference, particularly in optics.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Fresnel integrals canonical | 3 |
| Fresnel cosine integral | 2 |
| Fresnel sine integral | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T1783970 — 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: Fresnel integrals Context triple: [Fresnel equations, relatedTo, Fresnel integrals]
-
A.
Fraunhofer diffraction
Fraunhofer diffraction is the far-field diffraction pattern of waves, typically light, observed when both the source and observation screen are effectively at infinite distance or made so with lenses, producing characteristic interference patterns.
-
B.
Jacobi elliptic functions
Jacobi elliptic functions are a family of doubly periodic complex functions that generalize trigonometric functions and play a central role in the theory of elliptic integrals and many areas of mathematical physics.
-
C.
Fresnel equations
The Fresnel equations are fundamental formulas in optics that describe how light is partially reflected and transmitted at the boundary between two media with different refractive indices, depending on polarization and angle of incidence.
-
D.
Gaussian integral
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.
-
E.
Fresnel zones
Fresnel zones are concentric regions on a wavefront used in wave optics to analyze and predict diffraction and interference effects, especially in near-field conditions.
- 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: Fresnel integrals Target entity description: Fresnel integrals are special functions in mathematics that describe the complex oscillatory behavior of wave diffraction and interference, particularly in optics.
-
A.
Fraunhofer diffraction
Fraunhofer diffraction is the far-field diffraction pattern of waves, typically light, observed when both the source and observation screen are effectively at infinite distance or made so with lenses, producing characteristic interference patterns.
-
B.
Jacobi elliptic functions
Jacobi elliptic functions are a family of doubly periodic complex functions that generalize trigonometric functions and play a central role in the theory of elliptic integrals and many areas of mathematical physics.
-
C.
Fresnel equations
The Fresnel equations are fundamental formulas in optics that describe how light is partially reflected and transmitted at the boundary between two media with different refractive indices, depending on polarization and angle of incidence.
-
D.
Gaussian integral
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.
-
E.
Fresnel zones
Fresnel zones are concentric regions on a wavefront used in wave optics to analyze and predict diffraction and interference effects, especially in near-field conditions.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical function
ⓘ
plane curve ⓘ special function ⓘ special function ⓘ special function ⓘ |
| appearIn |
Kirchhoff diffraction theory
ⓘ
surface form:
Kirchhoff diffraction formula
scalar diffraction theory ⓘ |
| behaviorAtInfinity |
C(∞) = 1/2
ⓘ
S(∞) = 1/2 ⓘ |
| behaviorAtMinusInfinity |
C(-∞) = -1/2
ⓘ
S(-∞) = -1/2 ⓘ |
| definedByIntegral |
C(x) = ∫₀ˣ cos(π t² / 2) dt
ⓘ
S(x) = ∫₀ˣ sin(π t² / 2) dt ⓘ |
| definedUsing |
Fresnel integrals
self-linksurface differs
ⓘ
surface form:
Fresnel cosine integral
Fresnel integrals self-linksurface differs ⓘ
surface form:
Fresnel sine integral
|
| extendTo | complex variable ⓘ |
| field |
mathematical analysis
ⓘ
optics ⓘ wave physics ⓘ |
| hasComponent |
Fresnel integrals
self-linksurface differs
ⓘ
surface form:
Fresnel cosine integral
Fresnel integrals self-linksurface differs ⓘ
surface form:
Fresnel sine integral
|
| namedAfter | Augustin-Jean Fresnel ⓘ |
| property |
bounded for real arguments
ⓘ
non-elementary functions ⓘ oscillatory integrals ⓘ |
| relatedTo |
Cornu spiral
ⓘ
Fourier transform methods in optics ⓘ Gaussian integral ⓘ
surface form:
Gaussian integrals
error function ⓘ stationary phase method ⓘ |
| satisfy | second-order linear differential equations ⓘ |
| symbol |
C(x)
ⓘ
S(x) ⓘ |
| usedFor |
analysis of circular aperture diffraction
ⓘ
analysis of slit diffraction ⓘ approximating diffraction patterns ⓘ computing intensity distributions in optics ⓘ designing optical systems with apertures ⓘ modeling wavefront propagation ⓘ |
| usedIn |
Cornu spiral
ⓘ
Fresnel diffraction theory ⓘ
surface form:
Fresnel diffraction
Fresnel zone plate analysis ⓘ antenna radiation pattern analysis ⓘ diffraction theory ⓘ edge diffraction problems ⓘ interference theory ⓘ near-field diffraction ⓘ |
| variableDomain | real variable x ⓘ |
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: Fresnel integrals Description of subject: Fresnel integrals are special functions in mathematics that describe the complex oscillatory behavior of wave diffraction and interference, particularly in optics.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Fresnel sine integral
this entity surface form:
Fresnel cosine integral
subject surface form:
Cornu spiral
this entity surface form:
Fresnel sine integral
subject surface form:
Cornu spiral
this entity surface form:
Fresnel cosine integral