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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
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
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
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
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
} 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
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
Sequence of differential equation solutions
Bessel function J α ( x ) {\displaystyle J_{\alpha }(x)} . a m {\displaystyle a_{m}} is the m {\displaystyle m} -th zero of the Airy function Ai ( x
Laguerre_polynomials
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
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
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
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
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
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
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
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
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
Equations describing elastic deformation
conditions. The Airy stress function is a special case of the Maxwell stress functions, in which it is assumed that A=B=0 and C is a function of x and y only
Stress_functions
Response if 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
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
Description of limiting behavior of a function
{\frac {1}{4n{\sqrt {3}}}}e^{\pi {\sqrt {\frac {2n}{3}}}}} Airy function The Airy function, Ai(x), is a solution of the differential equation y″ − xy
Asymptotic_analysis
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
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
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
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
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
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
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
Solution of a confluent hypergeometric equation
z)=(1+z)e^{z}.} 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
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
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
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
Asymptotic values of Hermite or Laguerre polynomials
(-3^{-1/3}t)} and Ai {\displaystyle \operatorname {Ai} } denotes the Airy function. Let L n ( α ) ( x ) {\displaystyle L_{n}^{(\alpha )}(x)} denote the
Plancherel–Rotach_asymptotics
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
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
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
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
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
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
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
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
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
\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
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
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
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
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
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
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
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
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
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
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
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
and Airy functions, Constants and Units, Trigonometric functions, etc. Statistics Tools for basic and advanced statistics: Estimators, Gamma functions, Beta
Colt_(libraries)
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
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
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
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
Swimming venue in Hamburg, Germany
its kind. Reminiscent of a butterfly—an allegory to its function as a swimming venue – the airy roof structure earned the Alsterschwimmhalle its nickname
Alsterschwimmhalle
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)
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
Twelve 30° sectors of the ecliptic, as defined by Western astrology
volatility, speed, adaptability, and primarily centrifugal communicative function. Feminine polarity planets are associated with receptivity, consolidation
Astrological_sign
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
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
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
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
Ability of an imaging system to resolve detail
half-maximum (FWHM) of the point spread function is often used to avoid the difficulty of measuring the Airy disc. This, combined with the rastered illumination
Optical_resolution
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
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)
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
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
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
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
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
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
Design of interior spaces to benefit its occupants
the gaping ruins of torn-down buildings- the effect is light, delicate and airy. It is usually around the majlis, around the coffee hearth and along the
Interior_design
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
Capital and chief port of Jamaica
in rural St. Andrew such as Gordon Town, Mavis Bank, Lawrence Tavern, Mt. Airy, and Bull Bay would not be described as being in Kingston city. Two districts
Kingston,_Jamaica
Elevated base for storage buildings
often set on top of staddle stones to keep out predators and provide dry and airy conditions. Small staddle stones were used to support small roofed box-shaped
Staddle_stones
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
Clock tower in London, England
upon its completion. It was designed by Edmund Beckett Denison and George Airy, the Astronomer Royal, and constructed by Edward John Dent and Frederick
Big_Ben
Love focused on feelings
If we reason, we would be understood; if we imagine, we would that the airy children of our brain were born anew within another's; if we feel, we would
Romance
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
Most populous city in Pennsylvania, US
Irish-American population, is home to a Jewish and Russian population. Mount Airy in Northwest Philadelphia also contains a Jewish community. Nearby Chestnut
Philadelphia
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
2012 video game
game's title changes from Where The Fairy Flies/Flying Fairy to Airy Lies/Lying Airy. As depicted in a special video for Bravely Second: End Layer. ブレイブリーデフォルト
Bravely_Default
Non-linear stochastic partial differential equation
(1+1) KPZ universality class and govern the long time fluctuations are the Airy processes and the KPZ fixed point. Due to the nonlinearity in the equation
Kardar–Parisi–Zhang_equation
Elasticity equation
showed that the general solution can be expressed in terms of an Airy stress function of the form φ ( r , θ ) = A 0 r 2 + B 0 r 2 ln ( r ) + C 0 ln
Michell_solution
Transport of energy by wind waves, and the capture of that energy to do useful work
Under typical conditions, however, the movement of waves is described by Airy wave theory, which posits that fluid motion is roughly irrotational, pressure
Wave_power
AIRY FUNCTION
AIRY FUNCTION
Boy/Male
Spanish
Hairy.
Boy/Male
Hebrew
Hairy.
Boy/Male
Latin
Hairy.
Boy/Male
Latin
Hairy.
Girl/Female
Muslim
Fairy
Girl/Female
Tamil
Fairy
Girl/Female
American, Australian, Gujarati, Indian
Woman Known to have Magical Powers
Boy/Male
Latin
Hairy.
Boy/Male
Australian, Dutch, French, German, Greek
Superior; Best of Thinkers; Diminutive of Aristotle
Surname or Lastname
Americanized spelling of French Hary.English
Americanized spelling of French Hary.English : variant spelling of Airey.
Girl/Female
Indian
Fairy
Girl/Female
Muslim
Fairy
Girl/Female
Indian
The beginning, The principle, The breathe of life
Boy/Male
Latin
Hairy.
Boy/Male
Greek
Superior; best of thinkers. Famous Bearers: ancient Greek philosopher Aristotle, and Greek...
Girl/Female
Native American
Fairy.
Girl/Female
Tamil
Fairy
Boy/Male
Arabic
Lovely
Boy/Male
Gujarati, Indian, Marathi, Punjabi, Sikh
God Rama
Girl/Female
Muslim
Fairy
AIRY FUNCTION
AIRY FUNCTION
Boy/Male
Tamil
Lord of strength
Boy/Male
Muslim
Lord Shiva, Lord Sun or north-east direction, Desiring and wishing
Boy/Male
Arabic
Respected; Honour
Boy/Male
Bengali, Indian
Beautiful
Boy/Male
Hindu, Indian
Unconquerable; Invincible
Boy/Male
Tamil
Vedviyan | வேதà¯à®µà®¿à®¯à®¨
Boy/Male
Indian
Always Winner
Surname or Lastname
English
English : unexplained. perhaps a habitational name, from a lost or unidentified place, possibly in Worcestershire, where the surname is frequent.
Boy/Male
Tamil
Daughter, Goddess Durga, Great achiever, Happiness, Lord Shivas son, Young Man
Boy/Male
American, Australian, British, Chinese, Christian, English
Hugh's Son; Son of the Hooded Man
AIRY FUNCTION
AIRY FUNCTION
AIRY FUNCTION
AIRY FUNCTION
AIRY FUNCTION
n.
To expose to heat, for the purpose of expelling dampness, or of warming; as, to air linen; to air liquors.
a.
Drawn in air; imaginary.
a.
Given by fairies; as, fairy money.
a.
Relating or belonging to air; high in air; aerial; as, an airy flight.
a.
Slacked, or pulverized, by exposure to the air; as, air-slacked lime.
n.
To expose to the air for the purpose of cooling, refreshing, or purifying; to ventilate; as, to air a room.
n.
The peculiar look, appearance, and bearing of a person; mien; demeanor; as, the air of a youth; a heavy air; a lofty air.
a.
Capable of endurance; tough; sinewy; as, a wiry frame or constitution.
a.
Consisting of air; as, an airy substance; the airy parts of bodies.
a.
Having an affected manner; being in the habit of putting on airs; affectedly grand.
n.
An imaginary being inhabiting the air; a fairy.
n.
A particular state of the atmosphere, as respects heat, cold, moisture, etc., or as affecting the sensations; as, a smoky air, a damp air, the morning air, etc.
n.
Odoriferous or contaminated air.
a.
Resembling air; thin; unsubstantial; not material; airlike.
a.
Relating to the spirit or soul; delicate; graceful; as, airy music.
n.
Air in motion; a light breeze; a gentle wind.
n.
A dairy farm.
a.
Open to a free current of air; exposed to the air; breezy; as, an airy situation.
n.
Any aeriform body; a gas; as, oxygen was formerly called vital air.
a.
Abounding with deep mud; full of mire; muddy; as, a miry road.