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Special function in the physical sciences
mathematics, the Airy function (or Airy function of the first kind) A i ( x ) {\displaystyle \mathbf {Ai({\boldsymbol {x}})} } is a special function named after
Airy_function
the Airy zeta function, studied by Crandall (1996), is a function analogous to the Riemann zeta function and related to the zeros of the Airy function. The
Airy_zeta_function
Mathematical concept
cut) because these approximations contain multi-valued functions. In contrast, the Airy function is single valued and indeed entire and therefore, in order
Stokes_phenomenon
Topics referred to by the same term
Look up airy in Wiktionary, the free dictionary. Airy may refer to: Airy (surname) Airy (lunar crater) Airy (Martian crater) Airy-0, a smaller crater
Airy
Index of lists with the same name
} Zeta functions include: Airy zeta function, related to the zeros of the Airy function Arakawa–Kaneko zeta function Arithmetic zeta function Artin–Mazur
List_of_zeta_functions
Generalized function whose value is zero everywhere except at zero
the Euler–Tricomi equation of transonic gas dynamics, is the rescaled Airy function ε − 1 / 3 Ai ( x ε − 1 / 3 ) . {\displaystyle \varepsilon ^{-1/3}\operatorname
Dirac_delta_function
Diffraction pattern in optics
In optics, the Airy disk (or Airy disc) and Airy pattern are descriptions of the best-focused spot of light that a perfect lens with a circular aperture
Airy_disk
process with almost surely continuous sample paths. The Airy process is named after the Airy function. The process can be defined through its finite-dimensional
Airy_process
Conjecture in algebraic geometry
"Intersection theory on the moduli space of curves and the matrix Airy function", Communications in Mathematical Physics, 147 (1): 1–23, Bibcode:1992CMaPh
Witten_conjecture
Area of mathematics
and so the edge of the rainbow always has the shape of an Airy function. The same Airy function fold catastrophe can be seen in nuclear-nuclear scattering
Catastrophe_theory
Equations describing elastic deformation
functions: ∇4A+∇4B+∇4C=3(∂2A∂x2+∂2B∂y2+∂2C∂z2)/(2−ν), These must also yield a stress tensor which obeys the specified boundary conditions. The Airy stress
Stress_functions
Field of radiation
An Airy beam is a propagation invariant wave whose main intensity lobe propagates along a curved parabolic trajectory while being resilient to perturbations
Airy_beam
Function used in signal processing
product of two sinc functions vs. an Airy function, respectively. Conventions: w 0 ( x ) {\displaystyle w_{0}(x)} is a zero-phase function (symmetrical about
Window_function
Function that is holomorphic on the whole complex plane
equations are entire functions. For example, the exponential function, sine, cosine, Airy functions and Parabolic cylinder functions arise in this way.
Entire_function
English mathematician and astronomer (1801–1892)
Sir George Biddell Airy (/ˈɛəri/; 27 July 1801 – 2 January 1892) was an English mathematician and astronomer, as well as the Lucasian Professor of Mathematics
George_Biddell_Airy
Solution method for linear differential equations
differential equation is known as the Airy equation, and the solution may be written in terms of Airy functions, Ψ ( x ) = C A Ai ( U 1 3 ⋅ ( x − x
WKB_approximation
Airy functions Ai(x) and Bi(x) Airy points. Airy stress functions Airy transform Airy wave theory. Airy zeta function Airy–Heiskanen model, see "Airy hypothesis"
List of things named after George Airy
List_of_things_named_after_George_Airy
modular forms, which include J-invariant Dedekind eta function Airy function Bessel functions: Defined by a differential equation; useful in astronomy
List of mathematical functions
List_of_mathematical_functions
Sequence of differential equation solutions
where Jα is a Bessel function of the first kind. See also:. Let ν=4n+2α+2. Let Ai be the Airy function. Let α be arbitrary and real, ϵ and
Laguerre_polynomials
Response of an optical system to a point source of light
produces an Airy disc in the image plane. The argument of the function J1(x)/x is important, because this determines the scaling of the Airy disc (in other
Point_spread_function
Short "burst" or "envelope" of restricted wave action that travels as a unit
group velocity, and always disperses, there exists a wave function based on Airy functions, that propagates freely without envelope dispersion, maintaining
Wave_packet
Study of classical optics using Fourier transforms
"point spread function" viewpoint would be to ask how well a lens transforms an Airy function in the object plane into an Airy function in the image plane
Fourier_optics
Polynomial sequence
would be turned back. This is a fold catastrophe, at which point the Airy function is needed. A better approximation, which accounts for the variation
Hermite_polynomials
Description of limiting behavior of a function
where the order of addends is not considered. p(n)∼14n3eπ2n3 Airy function The Airy function, Ai(x), is a solution of the differential equation y″ − xy
Asymptotic_analysis
Optical device with parallel mirrors
resolved. Therefore, the Airy distribution becomes the underlying fundamental function and the measurement delivers a sum of Airy distributions. The parameters
Fabry–Pérot_interferometer
Quantum mechanical model
classical explanation, but can be modeled using an Airy function. Using properties of the Airy function, one may estimate the probability of finding the
Quantum_harmonic_oscillator
Quantum mechanical phenomenon
solution to connect them. Hence, the Airy function solutions will asymptote into sine, cosine and exponential functions in the proper limits. The relationships
Quantum_tunnelling
Series of functions in mathematics
occurred when George Stokes studied the Airy function. He discovered that although the Airy function is an entire function, it requires different asymptotic
Asymptotic_expansion
Probability distribution of random variable
(i(2c^{2})^{-1/3}s)}},\ \ \ s\in \mathbf {R} } and where Ai is the Airy function. Thus fc is symmetric about 0 and the density ƒZ = ƒ1. Groeneboom (1989)
Chernoff's_distribution
Mathematical function having a characteristic S-shaped curve or sigmoid curve
sigmoid function is any mathematical function whose graph has a characteristic S-shaped or sigmoid curve. A common example of a sigmoid function is the
Sigmoid_function
Abel: Abel polynomials - Abelian function - Abel–Gontscharoff interpolating polynomial Sir George Biddell Airy: Airy function Waleed Al-Salam (1926–1996):
List of eponyms of special functions
List_of_eponyms_of_special_functions
Change in optical absorption by a semiconductor when an electric field is applied
applied, the electron and hole wavefunctions become Airy functions rather than plane waves. The Airy function includes a "tail" which extends into the classically
Franz–Keldysh_effect
Proportion of observed intensity to ideal intensity, in astronomical imaging
point spread function (PSF) which describes the image formed from an object with no spatial extent (a "point source"), is given by the Airy disk as illustrated
Strehl_ratio
Form of error in digital signals; spurious signals near sharp transitions
Fraunhofer diffraction yields the Airy disk as point spread function, which has a ringing pattern. The Bessel function of the first kind, J 0 , {\displaystyle
Ringing_artifacts
Mathematical constants
ISSN 1572-9303. "Airy and Related Function". dlmf.nist.gov. Retrieved 9 October 2022. Weisstein, Eric W. "Airy Zeta Function". MathWorld. Weisstein
Particular values of the gamma function
Particular_values_of_the_gamma_function
Mathematical function
neural networks. In fluorescence microscopy a 2D Gaussian function is used to approximate the Airy disk, describing the intensity distribution produced by
Gaussian_function
almost everywhere. AFSOC - Assume for the sake of contradiction Ai – Airy function. AL – Action limit. Alt – alternating group (Alt(n) is also written
List of mathematical abbreviations
List_of_mathematical_abbreviations
Russian and French mathematician (born 1964)
"Intersection theory on the moduli space of curves and the matrix Airy function", Communications in Mathematical Physics, 147 (1): 1–23, Bibcode:1992CMaPh
Maxim_Kontsevich
Stochastic point process in mathematics
polynomial. The Airy process is governed by the so called extended Airy kernel which is a generalization of the Airy kernel function K A i ( x , y ) =
Determinantal_point_process
Asymptotic values of Hermite or Laguerre polynomials
(-3^{-1/3}t)} and Ai {\displaystyle \operatorname {Ai} } denotes the Airy function. Let He n ( x ) {\displaystyle \operatorname {He} _{n}(x)} denote
Plancherel–Rotach_asymptotics
Oscillating dynamical system with nonlinear damping
where α ≈ 2.338 is the smallest root of Ai(–α) = 0, where Ai is the Airy function. (Section 9.7) ( contains a derivation, but has a misprint of 3α to
Van_der_Pol_oscillator
Analytic function with prescribed zeros
|k-\alpha _{0}^{\pm 1}n|=O(n^{1/3})} , the transition is described by the Airy function, and the largest coefficients are of order n − 1 / 3 {\displaystyle
Blaschke_product
Solution of a confluent hypergeometric equation
z)=(1+z)ez. Bateman's function Bessel functions and many related functions such as Airy functions, Kelvin functions, Hankel functions. For example, in the
Confluent hypergeometric function
Confluent_hypergeometric_function
Stochastic process
{\displaystyle a_{j}} are the zeros of the Airy function and U {\displaystyle U} is the confluent hypergeometric function. Janson and Louchard (2007) show that
Brownian_excursion
Asymptotic analysis used when integrating rapidly-varying complex exponentials
oscillatory integrals for Morse functions. The degenerate case requires further techniques (see for example Airy function). The essential statement is this
Stationary phase approximation
Stationary_phase_approximation
Type of electromagnetic diffraction
understanding and calculation of creeping waves. They are described by Airy functions. Neubauer, W. G.; Uginčius, P.; Überall, H. (1969-05-01). "Theory of
Creeping_wave
problem, many-body problem Ballistics Airy function Bessel function Legendre polynomials Hypergeometric function Angular velocity Angular momentum Angular
List of dynamical systems and differential equations topics
List_of_dynamical_systems_and_differential_equations_topics
equations List of named differential equations Vallée, Olivier (2004). Airy functions and applications to physics. Internet Archive. Hackensack, NJ : World
List of linear ordinary differential equations
List_of_linear_ordinary_differential_equations
Quantum variations of random walks
like the Hermite functions which (1) asymptotically oscillate in the classically allowed region, (2) is approximated by the Airy function around the wall
Quantum_walk
Computation of an antiderivatives
special functions such as Airy function, error function, Bessel functions, and all hypergeometric functions. A fundamental property of holonomic functions is
Symbolic_integration
Set of nonlinear partial differential equations
Föppl–von Kármán equations can be reduced to two by introducing the Airy stress function φ {\displaystyle \varphi } where σ 11 = ∂ 2 φ ∂ x 2 2 , σ 22
Föppl–von_Kármán_equations
Signal-processing operation
transmission, or a mechanical structure. In optics, it is primarily used to remove Airy disks caused by diffraction around an intensity peak, improving the focus
Apodization
Town in Maryland
Mount Airy is a town in Carroll and Frederick counties in the U.S. state of Maryland. It is part of both the Washington Metropolitan Area and Baltimore
Mount_Airy,_Maryland
Fluid dynamics theory on gravity waves
In fluid dynamics, Airy wave theory (often referred to as linear wave theory) gives a linearised description of the propagation of gravity waves on the
Airy_wave_theory
Study of Galois symmetry groups of differential fields
differential equation are elementary functions (trigonometric functions in this case). The differential Galois group of the Airy equation, y ″ − x y = 0 {\displaystyle
Differential_Galois_theory
Type of functions, in mathematical analysis
Airy functions Ai ( x ) {\displaystyle \operatorname {Ai} (x)} , Bi ( x ) {\displaystyle \operatorname {Bi} (x)} The class of holonomic functions
Holonomic_function
caustic. Near a generic fold caustic, the local model is given by the Airy function. Its oscillatory asymptotics, Ai ( − z ) ∼ 1 π z 1 / 4 sin ( 2 3 z
Maslov_index
and Airy functions, Constants and Units, Trigonometric functions, etc. Statistics Tools for basic and advanced statistics: Estimators, Gamma functions, Beta
Colt_(libraries)
Infinite sequence of differential equations
generating function is a tau-function of the KdV hierarchy satisfying the string equation, and Maxim Kontsevich proved the conjecture using the matrix Airy function
Korteweg–De_Vries_hierarchy
Mathematical transform that expresses a function of time as a function of frequency
takes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output
Fourier_transform
2-dimensional polar coordinate function
A sombrero function (sometimes called besinc function or jinc function) is the 2-dimensional polar coordinate analog of the sinc function, and is so-called
Sombrero_function
Eigenvalue equation
,} where A i ( ⋅ ) {\displaystyle Ai\left(\cdot \right)} is the Airy function of the first kind. Substitution of the superposition solution φ = ∑
Orr–Sommerfeld_equation
Probability distribution
A_{s}} ("Airy kernel") on square integrable functions on the half line ( s , ∞ ) {\displaystyle (s,\infty )} , given in terms of Airy functions Ai by A
Tracy–Widom_distribution
\left(-{\frac {t^{3}}{3}}+xt\right)\,dt.} The Scorer's functions can also be defined in terms of Airy functions: G i ( x ) = B i ( x ) ∫ x ∞ A i ( t ) d t + A
Scorer's_function
Special functions in mathematics
terms of special functions: rational solutions for 2, 3, 4, 5, 6; algebraic solutions for 3, 5, 6; other special-function solutions: Airy for 2, Bessel for
Painlevé_transcendents
Probability distribution in physics
\end{aligned}}} where B i {\displaystyle \mathrm {Bi} } is the Airy function of the second kind and B i ′ {\displaystyle \mathrm {Bi} '} its derivative
Holtsmark_distribution
Ability of any image-forming device to distinguish small details of an object
central Airy disc of the diffraction pattern. This number is more precisely 1.21966989... (OEIS: A245461), the first zero of the order-one Bessel function of
Angular_resolution
Technique to find asymptotic expansions
the method and one can see the integral representation of the complex Airy function. Chester–Friedman–Ursell note to write h ( α , w ) {\displaystyle h(\alpha
Method of Chester–Friedman–Ursell
Method_of_Chester–Friedman–Ursell
Historic church in Pennsylvania, United States
Grace Church, Mt. Airy (Grace Epiphany Church) is an historic Episcopal church, which is located at 224 East Gowen Avenue in the Mount Airy neighborhood of
Grace_Church,_Mt._Airy
Computer graphics effect
the bright light appears to bleed beyond its natural borders. The Airy disc function falls off very quickly but has very wide tails (actually, infinitely
Bloom_(shader_effect)
Mathematical function in general imaging
simulations of HRTEM images. Airy disk, different but similar phenomena in light Optical transfer function Point spread function Transmission electron microscopy
Contrast_transfer_function
King of England, Scotland, and Ireland from 1660 to 1685
367–374; Miller 1991, pp. 306–309. Airy 1904, p. 7. Uglow 2009, p. 220; Fraser 1979, p. 23 Falkus 1972, p. 17; Airy 1904, p. 9 Carvalho, Cristina (2014)
Charles_II_of_England
American mathematician (1915–1958)
}}+{\frac {L(L+1)}{\rho ^{2}}})y=0} . By expressing them in terms of the Airy functions as y 1 ( x , μ ) = ϕ ( x , μ ) A i ( x ) + ψ ( x , μ ) A i ′ ( x ) {\displaystyle
Milton_Abramowitz
Distance over which a wave's shape repeats
physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it
Wavelength
Visual aura associated with migraine
a visual aura that was first described by 19th-century physician Hubert Airy (1838–1903). Originating from the brain, it may precede a migraine headache
Scintillating_scotoma
Clock outside Greenwich Observatory, London
new transatlantic submarine cable. Airy's report to the Observatory's Board of Visitors in 1853 explained the function of the Shepherd motor clock: This
Shepherd_Gate_Clock
Dutch statistician
Groeneboom, Piet (1989). "Brownian motion with a parabolic drift and Airy functions". Probability Theory and Related Fields. 81 (1): 79–109. doi:10.1007/BF00343738
Piet_Groeneboom
Interference phenomenon of waves
circular aperture is known as the Airy disk. The Airy disk has the following intensity distribution as a function of angle θ: I ( θ ) = I 0 ( 2 J 1 (
Diffraction
Excavated subterranean space around a building, for lighting a basement
house descriptions, the area was usually called the "airy", which suggests that its primary function was ventilation, needed to prevent cooking smells from
Area_(architecture)
Ancient Greek chthonic goddess
from Persephone's fury. This specter drives mortals to madness with her airy apparitions as she appears in weird shapes and strange forms, now plain to
Melinoë
Observatory in London, England
Positional astronomy was one of the primary functions of Greenwich for the Admiralty. Astronomer Royal Airy was an advocate of this and the transit circle
Royal_Observatory,_Greenwich
Far-field diffraction
the Airy diffraction pattern. It can be seen that most of the light is in the central disk. The angle subtended by this disk, known as the Airy disk
Fraunhofer_diffraction
Optical system with resolution performance at the instrument's theoretical limit
spread functions (PSF). The point spread function of a diffraction limited circular-aperture lens is simply the Airy disk. The point spread function of the
Diffraction-limited_system
Difference between logarithm and harmonic series
"DLMF: §9.12 Scorer Functions ‣ Related Functions ‣ Chapter 9 Airy and Related Functions". dlmf.nist.gov. Retrieved 2024-11-01. Rosser, J. Barkley; Schoenfeld
Euler's_constant
Optical phenomenon of the sky
rather than small ice crystals. The corona intensity pattern resembles an Airy disk, since the diffraction pattern of a round obstacle is identical to that
Corona_(optical_phenomenon)
Largest integer that divides given integers
by the same integer). Barlow, Peter; Peacock, George; Lardner, Dionysius; Airy, Sir George Biddell; Hamilton, H. P.; Levy, A.; De Morgan, Augustus; Mosley
Greatest_common_divisor
Adolf Hitler's Bavarian holiday home
designer, and furnisher, as well as architect", and the chalet as "bright and airy" with "a light jade-green colour scheme"; caged Harz Roller canaries were
Berghof_(residence)
American singer-songwriter (born 1948)
29, 2013. Retrieved September 30, 2012. Booth Moore (October 23, 1997). "Airy Godmother". Los Angeles Times. Archived from the original on May 9, 2009
Stevie_Nicks
British physicist, engineer and mathematician (1824–1907)
involved much detailed work to develop principles identified by George Biddell Airy and others, but contributed little in terms of novel physical thinking. Thomson's
Lord_Kelvin
Quality of a musical note or sound or tone
different parts of the string. E.g., playing sul tasto produces a light, airy timbre, whereas sul ponticello produces a harsh, even, aggressive timbre
Timbre
English mathematician, philosopher, and engineer (1791–1871)
Sciences in 1832. Babbage was out of sympathy with colleagues: George Biddell Airy, his predecessor as Lucasian Professor of Mathematics at Trinity College
Charles_Babbage
Radius which encloses 50% of the total light of a galaxy
{\displaystyle 2000\cdot I_{e}} . Airy disk Elliptical galaxy Globular cluster Open cluster Point spread function Sérsic profile "Half-light Radius"
Galaxy_effective_radius
Computer algebra system
x, 0)); ∑i=0∞(−1)ix2i(2i)! bessel_j(0, 4.5); −0.3205425089851214 airy_ai(1.5); 0.07174949700810543 Free and open-source software portal Mathematics
Maxima_(software)
and crew were killed. September 3 – Aeroflot Flight Sh-4 crashed on Mount Airy-Tash, while flying from Fruzne to Dushnabe. All 21 people on board were killed
List of accidents and incidents involving commercial aircraft
List_of_accidents_and_incidents_involving_commercial_aircraft
Protein filament that grows from follicles found in the dermis, or skin
density of Afro-hair, combined with its springy coils actually results in an airy, almost sponge-like structure that in turn, Jablonski argues, more likely
Hair
Mechanism for regulating the speed of clocks
Kew Observatory, UK. Airy's coal pit experiments: Starting in 1826, using methods similar to Bouguer, British astronomer George Airy attempted to determine
Pendulum
Acoustics Acousto-optic effect Acousto-optic modulator Acousto-optics Airy disc Airy wave theory Alfvén wave Alpha waves Amphidromic point Amplitude Amplitude
Index_of_wave_articles
Flour-based baked sweet
flavor; cakes made with butter tend to be more flavorful and have a lighter, airier texture. Varieties include: Butter cake (high butter content) Pound cake
Cake
Mechanism for regulating the speed of clocks
small increase in the period of swing with amplitude. In 1826 George Biddell Airy showed that for maximum isochronism the best place in its cycle to apply
Escapement
Supernatural being originating in folklore
widespread belief concerning ghosts is that they are composed of a misty, airy, or subtle material. Anthropologists link this idea to early beliefs that
Ghost
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