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36-484: [REDACTED] Look up airy in Wiktionary, the free dictionary. Airy may refer to: Sir George Biddell Airy (1801–1892), British Astronomer Royal from 1835 to 1881, for whom the following features, phenomena, and theories are named: Airy (lunar crater) Airy (Martian crater) Airy-0 , a smaller crater within the previous one on Mars, and which defines

72-4261: A formula for Bi( z ) when π /3 < | arg( z ) | < π or, equivalently, for Ai(− z ) and Bi(− z ) when | arg( z ) | < 2 π /3 but not zero, are: Ai ⁡ ( − z ) ∼   1 π z 1 / 4 sin ⁡ ( 2 3 z 3 / 2 + π 4 ) [ ∑ n = 0 ∞ ( − 1 ) n Γ ( 2 n + 5 6 ) Γ ( 2 n + 1 6 ) ( 3 4 ) 2 n 2 π ( 2 n ) ! z 3 n ] − 1 π z 1 / 4 cos ⁡ ( 2 3 z 3 / 2 + π 4 ) [ ∑ n = 0 ∞ ( − 1 ) n Γ ( 2 n + 11 6 ) Γ ( 2 n + 7 6 ) ( 3 4 ) 2 n + 1 2 π ( 2 n + 1 ) ! z 3 n + 3 / 2 ] Bi ⁡ ( − z ) ∼ 1 π z 1 / 4 cos ⁡ ( 2 3 z 3 / 2 + π 4 ) [ ∑ n = 0 ∞ ( − 1 ) n Γ ( 2 n + 5 6 ) Γ ( 2 n + 1 6 ) ( 3 4 ) 2 n 2 π ( 2 n ) ! z 3 n ] + 1 π z 1 4 sin ⁡ ( 2 3 z 3 / 2 + π 4 ) [ ∑ n = 0 ∞ ( − 1 ) n Γ ( 2 n + 11 6 ) Γ ( 2 n + 7 6 ) ( 3 4 ) 2 n + 1 2 π ( 2 n + 1 ) ! z 3 n + 3 / 2 ] . {\displaystyle {\begin{aligned}\operatorname {Ai} (-z)\sim &{}\ {\frac {1}{{\sqrt {\pi }}\,z^{1/4}}}\sin \left({\frac {2}{3}}z^{3/2}+{\frac {\pi }{4}}\right)\left[\sum _{n=0}^{\infty }{\dfrac {(-1)^{n}\,\Gamma \!\left(2n+{\frac {5}{6}}\right)\,\Gamma \!\left(2n+{\frac {1}{6}}\right)\left({\frac {3}{4}}\right)^{2n}}{2\pi \,(2n)!\,z^{3n}}}\right]\\[6pt]&{}-{\frac {1}{{\sqrt {\pi }}\,z^{1/4}}}\cos \left({\frac {2}{3}}z^{3/2}+{\frac {\pi }{4}}\right)\left[\sum _{n=0}^{\infty }{\dfrac {(-1)^{n}\,\Gamma \!\left(2n+{\frac {11}{6}}\right)\,\Gamma \!\left(2n+{\frac {7}{6}}\right)\left({\frac {3}{4}}\right)^{2n+1}}{2\pi \,(2n+1)!\,z^{3n\,+\,3/2}}}\right]\\[6pt]\operatorname {Bi} (-z)\sim &{}{\frac {1}{{\sqrt {\pi }}\,z^{1/4}}}\cos \left({\frac {2}{3}}z^{3/2}+{\frac {\pi }{4}}\right)\left[\sum _{n=0}^{\infty }{\dfrac {(-1)^{n}\,\Gamma \!\left(2n+{\frac {5}{6}}\right)\,\Gamma \!\left(2n+{\frac {1}{6}}\right)\left({\frac {3}{4}}\right)^{2n}}{2\pi \,(2n)!\,z^{3n}}}\right]\\[6pt]&{}+{\frac {1}{{\sqrt {\pi }}\,z^{\frac {1}{4}}}}\sin \left({\frac {2}{3}}z^{3/2}+{\frac {\pi }{4}}\right)\left[\sum _{n=0}^{\infty }{\dfrac {(-1)^{n}\,\Gamma \!\left(2n+{\frac {11}{6}}\right)\,\Gamma \!\left(2n+{\frac {7}{6}}\right)\left({\frac {3}{4}}\right)^{2n+1}}{2\pi \,(2n+1)!\,z^{3n\,+\,3/2}}}\right].\end{aligned}}} When | arg( z ) | = 0 these are good approximations but are not asymptotic because

108-507: A straight line as is the case in symmetric beams. This is at expense of the low-intensity tail being spread in the opposite direction, so the overall momentum of the beam is of course conserved. The Airy function underlies the form of the intensity near an optical directional caustic , such as that of the rainbow (called supernumerary rainbow). Historically, this was the mathematical problem that led Airy to develop this special function. In 1841, William Hallowes Miller experimentally measured

144-1027: A turning point (a point where the character of the solutions changes from oscillatory to exponential). For real values of x , the Airy function of the first kind can be defined by the improper Riemann integral : Ai ⁡ ( x ) = 1 π ∫ 0 ∞ cos ⁡ ( t 3 3 + x t ) d t ≡ 1 π lim b → ∞ ∫ 0 b cos ⁡ ( t 3 3 + x t ) d t , {\displaystyle \operatorname {Ai} (x)={\dfrac {1}{\pi }}\int _{0}^{\infty }\cos \left({\dfrac {t^{3}}{3}}+xt\right)\,dt\equiv {\dfrac {1}{\pi }}\lim _{b\to \infty }\int _{0}^{b}\cos \left({\dfrac {t^{3}}{3}}+xt\right)\,dt,} which converges by Dirichlet's test . For any real number x there

180-531: A turning point in the WKB approximation , when the potential may be locally approximated by a linear function of position. The triangular potential well solution is directly relevant for the understanding of electrons trapped in semiconductor heterojunctions . A transversally asymmetric optical beam, where the electric field profile is given by the Airy function, has the interesting property that its maximum intensity accelerates towards one side instead of propagating in

216-593: Is a positive real number M such that function t 3 3 + x t {\textstyle {\tfrac {t^{3}}{3}}+xt} is increasing, unbounded and convex with continuous and unbounded derivative on interval [ M , ∞ ) . {\displaystyle [M,\infty ).} The convergence of the integral on this interval can be proven by Dirichlet's test after substitution u = t 3 3 + x t . {\textstyle u={\tfrac {t^{3}}{3}}+xt.} y = Ai( x ) satisfies

252-996: Is a similar one for Bi( z ) , but only applicable when | arg( z ) | < π /3 : Bi ⁡ ( z ) ∼ 1 π z 1 / 4 exp ⁡ ( 2 3 z 3 / 2 ) [ ∑ n = 0 ∞ Γ ( n + 5 6 ) Γ ( n + 1 6 ) ( 3 4 ) n 2 π n ! z 3 n / 2 ] . {\displaystyle \operatorname {Bi} (z)\sim {\frac {1}{{\sqrt {\pi }}\,z^{1/4}}}\exp \left({\frac {2}{3}}z^{3/2}\right)\left[\sum _{n=0}^{\infty }{\dfrac {\Gamma \!\left(n+{\frac {5}{6}}\right)\,\Gamma \!\left(n+{\frac {1}{6}}\right)\left({\frac {3}{4}}\right)^{n}}{2\pi \,n!\,z^{3n/2}}}\right].} A more accurate formula for Ai( z ) and

288-413: Is taken and x is bounded away from the negative real axis. The formula for Bi( x ) is valid provided x is in the sector x ∈ C : | arg ⁡ ( x ) | < π 3 − δ {\displaystyle x\in \mathbb {C} :\left|\arg(x)\right|<{\tfrac {\pi }{3}}-\delta } for some positive δ. Finally,

324-512: The Gamma function . It follows that the Wronskian of Ai( x ) and Bi( x ) is 1/ π . When x is positive, Ai( x ) is positive, convex , and decreasing exponentially to zero, while Bi( x ) is positive, convex, and increasing exponentially. When x is negative, Ai( x ) and Bi( x ) oscillate around zero with ever-increasing frequency and ever-decreasing amplitude. This is supported by

360-416: The Airy equation y ″ − x y = 0. {\displaystyle y''-xy=0.} This equation has two linearly independent solutions. Up to scalar multiplication, Ai( x ) is the solution subject to the condition y → 0 as x → ∞ . The standard choice for the other solution is the Airy function of the second kind, denoted Bi( x ). It is defined as the solution with

396-1525: The Airy function are related to the Bessel functions : Ai ⁡ ( − x ) = x 9 [ J 1 / 3 ( 2 3 x 3 / 2 ) + J − 1 / 3 ( 2 3 x 3 / 2 ) ] , Bi ⁡ ( − x ) = x 3 [ J − 1 / 3 ( 2 3 x 3 / 2 ) − J 1 / 3 ( 2 3 x 3 / 2 ) ] . {\displaystyle {\begin{aligned}\operatorname {Ai} (-x)&{}={\sqrt {\frac {x}{9}}}\left[J_{1/3}\!\left({\frac {2}{3}}x^{3/2}\right)+J_{-1/3}\!\left({\frac {2}{3}}x^{3/2}\right)\right],\\\operatorname {Bi} (-x)&{}={\sqrt {\frac {x}{3}}}\left[J_{-1/3}\!\left({\frac {2}{3}}x^{3/2}\right)-J_{1/3}\!\left({\frac {2}{3}}x^{3/2}\right)\right].\end{aligned}}} Here, J ±1/3 are solutions of x 2 y ″ + x y ′ + ( x 2 − 1 9 ) y = 0. {\displaystyle x^{2}y''+xy'+\left(x^{2}-{\frac {1}{9}}\right)y=0.} The Scorer's functions Hi( x ) and -Gi( x ) solve

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432-554: The Airy function is A i ′ ⁡ ( x ) = − x π 3 K 2 / 3 ( 2 3 x 3 / 2 ) . {\displaystyle \operatorname {Ai'} (x)=-{\frac {x}{\pi {\sqrt {3}}}}\,K_{2/3}\!\left({\frac {2}{3}}x^{3/2}\right).} Functions K 1/3 and K 2/3 can be represented in terms of rapidly convergent integrals (see also modified Bessel functions ) For negative arguments,

468-1289: The Airy functions are related to the modified Bessel functions : Ai ⁡ ( x ) = 1 π x 3 K 1 / 3 ( 2 3 x 3 / 2 ) , Bi ⁡ ( x ) = x 3 [ I 1 / 3 ( 2 3 x 3 / 2 ) + I − 1 / 3 ( 2 3 x 3 / 2 ) ] . {\displaystyle {\begin{aligned}\operatorname {Ai} (x)&{}={\frac {1}{\pi }}{\sqrt {\frac {x}{3}}}\,K_{1/3}\!\left({\frac {2}{3}}x^{3/2}\right),\\\operatorname {Bi} (x)&{}={\sqrt {\frac {x}{3}}}\left[I_{1/3}\!\left({\frac {2}{3}}x^{3/2}\right)+I_{-1/3}\!\left({\frac {2}{3}}x^{3/2}\right)\right].\end{aligned}}} Here, I ±1/3 and K 1/3 are solutions of x 2 y ″ + x y ′ − ( x 2 + 1 9 ) y = 0. {\displaystyle x^{2}y''+xy'-\left(x^{2}+{\tfrac {1}{9}}\right)y=0.} The first derivative of

504-2243: The Airy functions can be extended to the complex plane, giving entire functions . The asymptotic behaviour of the Airy functions as | z | goes to infinity at a constant value of arg ( z ) depends on arg( z ) : this is called the Stokes phenomenon . For | arg( z ) | < π we have the following asymptotic formula for Ai( z ) : Ai ⁡ ( z ) ∼ 1 2 π z 1 / 4 exp ⁡ ( − 2 3 z 3 / 2 ) [ ∑ n = 0 ∞ ( − 1 ) n Γ ( n + 5 6 ) Γ ( n + 1 6 ) ( 3 4 ) n 2 π n ! z 3 n / 2 ] . {\displaystyle \operatorname {Ai} (z)\sim {\dfrac {1}{2{\sqrt {\pi }}\,z^{1/4}}}\exp \left(-{\frac {2}{3}}z^{3/2}\right)\left[\sum _{n=0}^{\infty }{\dfrac {(-1)^{n}\,\Gamma \!\left(n+{\frac {5}{6}}\right)\,\Gamma \!\left(n+{\frac {1}{6}}\right)\left({\frac {3}{4}}\right)^{n}}{2\pi \,n!\,z^{3n/2}}}\right].} or Ai ⁡ ( z ) ∼ e − ζ 4 π 3 / 2 z 1 / 4 [ ∑ n = 0 ∞ Γ ( n + 5 6 ) Γ ( n + 1 6 ) n ! ( − 2 ζ ) n ] . {\displaystyle \operatorname {Ai} (z)\sim {\dfrac {e^{-\zeta }}{4\pi ^{3/2}\,z^{1/4}}}\left[\sum _{n=0}^{\infty }{\dfrac {\Gamma \!\left(n+{\frac {5}{6}}\right)\,\Gamma \!\left(n+{\frac {1}{6}}\right)}{n!(-2\zeta )^{n}}}\right].} where ζ = 2 3 z 3 / 2 . {\displaystyle \zeta ={\tfrac {2}{3}}z^{3/2}.} In particular,

540-751: The Fourier transform of the Airy equation. Let y ^ = 1 2 π i ∫ y e − i k x d x {\textstyle {\hat {y}}={\frac {1}{2\pi i}}\int ye^{-ikx}dx} , then i y ^ ′ + k 2 y ^ = 0 {\displaystyle i{\hat {y}}'+k^{2}{\hat {y}}=0} , which then has solutions y ^ = C e i k 3 / 3 . {\displaystyle {\hat {y}}=Ce^{ik^{3}/3}.} There only one dimension of solutions because

576-454: The Fourier transform requires y to decay to zero fast enough, and Bi grows to infinity exponentially fast, so it cannot be obtained via Fourier transform. The Airy function is the solution to the time-independent Schrödinger equation for a particle confined within a triangular potential well and for a particle in a one-dimensional constant force field. For the same reason, it also serves to provide uniform semiclassical approximations near

612-527: The International Prototype Meter) Anna Airy (1882–1964), British artist Airy, a character in the video game Bravely Default Airy (musician) , South Korean indie pop musician See also [ edit ] Airey (disambiguation) Mount Airy (disambiguation) Aerie (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

648-428: The International Prototype Meter) Anna Airy (1882–1964), British artist Airy, a character in the video game Bravely Default Airy (musician) , South Korean indie pop musician See also [ edit ] Airey (disambiguation) Mount Airy (disambiguation) Aerie (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

684-464: The analog to supernumerary rainbow by shining light through a thin cylinder of water, then observing through a telescope. He observed up to 30 bands. In the mid-1980s, the Airy function was found to be intimately connected to Chernoff's distribution . The Airy function also appears in the definition of Tracy–Widom distribution which describes the law of largest eigenvalues in Random matrix . Due to

720-642: The asymptotic formulae below for the Airy functions. The Airy functions are orthogonal in the sense that ∫ − ∞ ∞ Ai ⁡ ( t + x ) Ai ⁡ ( t + y ) d t = δ ( x − y ) {\displaystyle \int _{-\infty }^{\infty }\operatorname {Ai} (t+x)\operatorname {Ai} (t+y)dt=\delta (x-y)} again using an improper Riemann integral. Neither Ai( x ) nor its derivative Ai'( x ) have positive real zeros. The "first" real zeros (i.e. nearest to x=0) are: As explained below,

756-634: The definition of the Airy function Ai( x ), it is straightforward to show its Fourier transform is given by F ( Ai ) ( k ) := ∫ − ∞ ∞ Ai ⁡ ( x )   e − 2 π i k x d x = e i 3 ( 2 π k ) 3 . {\displaystyle {\mathcal {F}}(\operatorname {Ai} )(k):=\int _{-\infty }^{\infty }\operatorname {Ai} (x)\ e^{-2\pi ikx}\,dx=e^{{\frac {i}{3}}(2\pi k)^{3}}.} This can be obtained by taking

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792-418: The definition of the Airy function to the complex plane by Ai ⁡ ( z ) = 1 2 π i ∫ C exp ⁡ ( t 3 3 − z t ) d t , {\displaystyle \operatorname {Ai} (z)={\frac {1}{2\pi i}}\int _{C}\exp \left({\tfrac {t^{3}}{3}}-zt\right)\,dt,} where

828-1164: The equation y ′′ − xy = 1/π . They can also be expressed in terms of the Airy functions: Gi ⁡ ( x ) = Bi ⁡ ( x ) ∫ x ∞ Ai ⁡ ( t ) d t + Ai ⁡ ( x ) ∫ 0 x Bi ⁡ ( t ) d t , Hi ⁡ ( x ) = Bi ⁡ ( x ) ∫ − ∞ x Ai ⁡ ( t ) d t − Ai ⁡ ( x ) ∫ − ∞ x Bi ⁡ ( t ) d t . {\displaystyle {\begin{aligned}\operatorname {Gi} (x)&{}=\operatorname {Bi} (x)\int _{x}^{\infty }\operatorname {Ai} (t)\,dt+\operatorname {Ai} (x)\int _{0}^{x}\operatorname {Bi} (t)\,dt,\\\operatorname {Hi} (x)&{}=\operatorname {Bi} (x)\int _{-\infty }^{x}\operatorname {Ai} (t)\,dt-\operatorname {Ai} (x)\int _{-\infty }^{x}\operatorname {Bi} (t)\,dt.\end{aligned}}} Using

864-572: The first few terms are Ai ⁡ ( z ) = e − ζ 2 π 1 / 2 z 1 / 4 ( 1 − 5 72 ζ + 385 10368 ζ 2 + O ( ζ − 3 ) ) {\displaystyle \operatorname {Ai} (z)={\frac {e^{-\zeta }}{2\pi ^{1/2}z^{1/4}}}\left(1-{\frac {5}{72\zeta }}+{\frac {385}{10368\zeta ^{2}}}+O(\zeta ^{-3})\right)} There

900-479: The formulae for Ai(− x ) and Bi(− x ) are valid if x is in the sector x ∈ C : | arg ⁡ ( x ) | < 2 π 3 − δ . {\displaystyle x\in \mathbb {C} :\left|\arg(x)\right|<{\tfrac {2\pi }{3}}-\delta .} It follows from the asymptotic behaviour of the Airy functions that both Ai( x ) and Bi( x ) have an infinity of zeros on

936-409: The 💕 [REDACTED] Look up airy in Wiktionary, the free dictionary. Airy may refer to: Sir George Biddell Airy (1801–1892), British Astronomer Royal from 1835 to 1881, for whom the following features, phenomena, and theories are named: Airy (lunar crater) Airy (Martian crater) Airy-0 , a smaller crater within the previous one on Mars, and which defines

972-397: The integral is over a path C starting at the point at infinity with argument − π /3 and ending at the point at infinity with argument π/3. Alternatively, we can use the differential equation y ′′ − xy = 0 to extend Ai( x ) and Bi( x ) to entire functions on the complex plane. The asymptotic formula for Ai( x ) is still valid in the complex plane if the principal value of x

1008-634: The intimate connection of random matrix theory with the Kardar–Parisi–Zhang equation , there are central processes constructed in KPZ such as the Airy process . The Airy function is named after the British astronomer and physicist George Biddell Airy (1801–1892), who encountered it in his early study of optics in physics (Airy 1838). The notation Ai( x ) was introduced by Harold Jeffreys . Airy had become

1044-476: The negative real axis. The function Ai( x ) has no other zeros in the complex plane, while the function Bi( x ) also has infinitely many zeros in the sector z ∈ C : π 3 < | arg ⁡ ( z ) | < π 2 . {\displaystyle z\in \mathbb {C} :{\tfrac {\pi }{3}}<\left|\arg(z)\right|<{\tfrac {\pi }{2}}.} For positive arguments,

1080-427: The prime meridian of the planet Airy wave theory , a linear theory describing the propagation of "gravity waves" on the surface of a fluid Airy disk , a diffraction pattern in optics Airy beam , a non-spreading, transversely accelerating optical wavepacket Airy function , a mathematical function Airy points , support points chosen to minimize the distortion of the length of a physical standard (such as

1116-427: The prime meridian of the planet Airy wave theory , a linear theory describing the propagation of "gravity waves" on the surface of a fluid Airy disk , a diffraction pattern in optics Airy beam , a non-spreading, transversely accelerating optical wavepacket Airy function , a mathematical function Airy points , support points chosen to minimize the distortion of the length of a physical standard (such as

Airy - Misplaced Pages Continue

1152-7000: The ratio between Ai(− z ) or Bi(− z ) and the above approximation goes to infinity whenever the sine or cosine goes to zero. Asymptotic expansions for these limits are also available. These are listed in (Abramowitz and Stegun, 1983) and (Olver, 1974). One is also able to obtain asymptotic expressions for the derivatives Ai'(z) and Bi'(z) . Similarly to before, when | arg( z ) | < π : Ai ′ ⁡ ( z ) ∼ − z 1 / 4 2 π exp ⁡ ( − 2 3 z 3 / 2 ) [ ∑ n = 0 ∞ 1 + 6 n 1 − 6 n ( − 1 ) n Γ ( n + 5 6 ) Γ ( n + 1 6 ) ( 3 4 ) n 2 π n ! z 3 n / 2 ] . {\displaystyle \operatorname {Ai} '(z)\sim -{\dfrac {z^{1/4}}{2{\sqrt {\pi }}\,}}\exp \left(-{\frac {2}{3}}z^{3/2}\right)\left[\sum _{n=0}^{\infty }{\frac {1+6n}{1-6n}}{\dfrac {(-1)^{n}\,\Gamma \!\left(n+{\frac {5}{6}}\right)\,\Gamma \!\left(n+{\frac {1}{6}}\right)\left({\frac {3}{4}}\right)^{n}}{2\pi \,n!\,z^{3n/2}}}\right].} When | arg( z ) | < π /3 we have: Bi ′ ⁡ ( z ) ∼ z 1 / 4 π exp ⁡ ( 2 3 z 3 / 2 ) [ ∑ n = 0 ∞ 1 + 6 n 1 − 6 n Γ ( n + 5 6 ) Γ ( n + 1 6 ) ( 3 4 ) n 2 π n ! z 3 n / 2 ] . {\displaystyle \operatorname {Bi} '(z)\sim {\frac {z^{1/4}}{{\sqrt {\pi }}\,}}\exp \left({\frac {2}{3}}z^{3/2}\right)\left[\sum _{n=0}^{\infty }{\frac {1+6n}{1-6n}}{\dfrac {\Gamma \!\left(n+{\frac {5}{6}}\right)\,\Gamma \!\left(n+{\frac {1}{6}}\right)\left({\frac {3}{4}}\right)^{n}}{2\pi \,n!\,z^{3n/2}}}\right].} Similarly, an expression for Ai'(− z ) and Bi'(− z ) when | arg( z ) | < 2 π /3 but not zero, are Ai ′ ⁡ ( − z ) ∼ − z 1 / 4 π cos ⁡ ( 2 3 z 3 / 2 + π 4 ) [ ∑ n = 0 ∞ 1 + 12 n 1 − 12 n ( − 1 ) n Γ ( 2 n + 5 6 ) Γ ( 2 n + 1 6 ) ( 3 4 ) 2 n 2 π ( 2 n ) ! z 3 n ] − z 1 / 4 π sin ⁡ ( 2 3 z 3 / 2 + π 4 ) [ ∑ n = 0 ∞ 7 + 12 n − 5 − 12 n ( − 1 ) n Γ ( 2 n + 11 6 ) Γ ( 2 n + 7 6 ) ( 3 4 ) 2 n + 1 2 π ( 2 n + 1 ) ! z 3 n + 3 / 2 ] Bi ′ ⁡ ( − z ) ∼   z 1 / 4 π sin ⁡ ( 2 3 z 3 / 2 + π 4 ) [ ∑ n = 0 ∞ 1 + 12 n 1 − 12 n ( − 1 ) n Γ ( 2 n + 5 6 ) Γ ( 2 n + 1 6 ) ( 3 4 ) 2 n 2 π ( 2 n ) ! z 3 n ] − z 1 / 4 π cos ⁡ ( 2 3 z 3 / 2 + π 4 ) [ ∑ n = 0 ∞ 7 + 12 n − 5 − 12 n ( − 1 ) n Γ ( 2 n + 11 6 ) Γ ( 2 n + 7 6 ) ( 3 4 ) 2 n + 1 2 π ( 2 n + 1 ) ! z 3 n + 3 / 2 ] {\displaystyle {\begin{aligned}\operatorname {Ai} '(-z)\sim &{}-{\frac {z^{1/4}}{{\sqrt {\pi }}\,}}\cos \left({\frac {2}{3}}z^{3/2}+{\frac {\pi }{4}}\right)\left[\sum _{n=0}^{\infty }{\frac {1+12n}{1-12n}}{\dfrac {(-1)^{n}\,\Gamma \!\left(2n+{\frac {5}{6}}\right)\,\Gamma \!\left(2n+{\frac {1}{6}}\right)\left({\frac {3}{4}}\right)^{2n}}{2\pi \,(2n)!\,z^{3n}}}\right]\\[6pt]&{}-{\frac {z^{1/4}}{{\sqrt {\pi }}\,}}\sin \left({\frac {2}{3}}z^{3/2}+{\frac {\pi }{4}}\right)\left[\sum _{n=0}^{\infty }{\frac {7+12n}{-5-12n}}{\dfrac {(-1)^{n}\,\Gamma \!\left(2n+{\frac {11}{6}}\right)\,\Gamma \!\left(2n+{\frac {7}{6}}\right)\left({\frac {3}{4}}\right)^{2n+1}}{2\pi \,(2n+1)!\,z^{3n\,+\,3/2}}}\right]\\[6pt]\operatorname {Bi} '(-z)\sim &{}\ {\frac {z^{1/4}}{{\sqrt {\pi }}\,}}\sin \left({\frac {2}{3}}z^{3/2}+{\frac {\pi }{4}}\right)\left[\sum _{n=0}^{\infty }{\frac {1+12n}{1-12n}}{\dfrac {(-1)^{n}\,\Gamma \!\left(2n+{\frac {5}{6}}\right)\,\Gamma \!\left(2n+{\frac {1}{6}}\right)\left({\frac {3}{4}}\right)^{2n}}{2\pi \,(2n)!\,z^{3n}}}\right]\\[6pt]&{}-{\frac {z^{1/4}}{{\sqrt {\pi }}\,}}\cos \left({\frac {2}{3}}z^{3/2}+{\frac {\pi }{4}}\right)\left[\sum _{n=0}^{\infty }{\frac {7+12n}{-5-12n}}{\dfrac {(-1)^{n}\,\Gamma \!\left(2n+{\frac {11}{6}}\right)\,\Gamma \!\left(2n+{\frac {7}{6}}\right)\left({\frac {3}{4}}\right)^{2n+1}}{2\pi \,(2n+1)!\,z^{3n\,+\,3/2}}}\right]\\\end{aligned}}} We can extend

1188-1878: The same amplitude of oscillation as Ai( x ) as x → −∞ which differs in phase by π /2 : Bi ⁡ ( x ) = 1 π ∫ 0 ∞ [ exp ⁡ ( − t 3 3 + x t ) + sin ⁡ ( t 3 3 + x t ) ] d t . {\displaystyle \operatorname {Bi} (x)={\frac {1}{\pi }}\int _{0}^{\infty }\left[\exp \left(-{\tfrac {t^{3}}{3}}+xt\right)+\sin \left({\tfrac {t^{3}}{3}}+xt\right)\,\right]dt.} The values of Ai( x ) and Bi( x ) and their derivatives at x = 0 are given by Ai ⁡ ( 0 ) = 1 3 2 / 3 Γ ( 2 3 ) , Ai ′ ⁡ ( 0 ) = − 1 3 1 / 3 Γ ( 1 3 ) , Bi ⁡ ( 0 ) = 1 3 1 / 6 Γ ( 2 3 ) , Bi ′ ⁡ ( 0 ) = 3 1 / 6 Γ ( 1 3 ) . {\displaystyle {\begin{aligned}\operatorname {Ai} (0)&{}={\frac {1}{3^{2/3}\,\Gamma \!\left({\frac {2}{3}}\right)}},&\quad \operatorname {Ai} '(0)&{}=-{\frac {1}{3^{1/3}\,\Gamma \!\left({\frac {1}{3}}\right)}},\\\operatorname {Bi} (0)&{}={\frac {1}{3^{1/6}\,\Gamma \!\left({\frac {2}{3}}\right)}},&\quad \operatorname {Bi} '(0)&{}={\frac {3^{1/6}}{\Gamma \!\left({\frac {1}{3}}\right)}}.\end{aligned}}} Here, Γ denotes

1224-448: The solution of the linear differential equation d 2 y d x 2 − k y = 0 {\displaystyle {\frac {d^{2}y}{dx^{2}}}-ky=0} is oscillatory for k <0 and exponential for k >0 , the Airy functions are oscillatory for x <0 and exponential for x >0 . In fact, the Airy equation is the simplest second-order linear differential equation with

1260-446: The title Airy . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Airy&oldid=1184029328 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages airy From Misplaced Pages,

1296-446: The title Airy . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Airy&oldid=1184029328 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Airy function Because

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