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 |
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 ⓘ |
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