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<title>Airy function</title>

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Airy function

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about the Airy special function. For the Airy stress function employed in solid mechanics, see <a href="Stress_functions" title="Stress functions">Stress functions</a>. For the Airy disk function that describes the optics diffraction pattern through a circular aperture, see <a href="Airy_disk" title="Airy disk">Airy disk</a>. For generic Airy distribution arising from optical resonance between two mirrors, see <a href="Fabry%E2%80%93P%C3%A9rot_interferometer" title="Fabry–Pérot interferometer">Fabry–Pérot interferometer</a>. For the Airy equation as an example of a linear dispersive partial differential equation, see <a href="Dispersive_partial_differential_equation" title="Dispersive partial differential equation">Dispersive partial differential equation</a>.</div>

In the physical sciences, the Airy function (or Airy function of the first kind) Ai(x) is a <a href="Special_function" class="mw-redirect" title="Special function">special function</a> named after the British astronomer <a href="George_Biddell_Airy" title="George Biddell Airy">George Biddell Airy</a> (1801–1892). The function Ai(x) and the related function Bi(x), are <a href="Linear_independence" title="Linear independence">linearly independent</a> solutions to the <a href="Differential_equation" title="Differential equation">differential equation</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}y}{dx^{2}}}-xy=0,}</annotation>

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known as the Airy equation or the Stokes equation.

Because the solution of the linear differential equation

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}y}{dx^{2}}}-ky=0}</annotation>

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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 <a href="Linear_differential_equation" title="Linear differential equation">linear differential equation</a> with a turning point (a point where the character of the solutions changes from oscillatory to exponential).

<div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2></div>

For real values of x, the Airy function of the first kind can be defined by the <a href="Improper_integral" title="Improper integral">improper</a> <a href="Riemann_integral" title="Riemann integral">Riemann integral</a>:

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<annotation encoding="application/x-tex">{\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,}</annotation>

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which converges by <a href="Dirichlet's_test" title="Dirichlet's test">Dirichlet's test</a>. For any <a href="Real_number" title="Real number">real number</a> x there is a positive real number M such that function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\tfrac {t^{3}}{3}}+xt}">

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<annotation encoding="application/x-tex">{\textstyle {\tfrac {t^{3}}{3}}+xt}</annotation>

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</math><img src="./2e22e21ba173e0f48a992fe70751b2b0403858cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.271ex; height:4.176ex;" alt="{\textstyle {\tfrac {t^{3}}{3}}+xt}" loading="lazy"> is increasing, unbounded and convex with continuous and unbounded derivative on interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [M,\infty ).}">

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<annotation encoding="application/x-tex">{\displaystyle [M,\infty ).}</annotation>

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</math><img src="./30dae9884089d06d8b11e27cbc346c1fe4efc44b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.998ex; height:2.843ex;" alt="{\displaystyle [M,\infty ).}" loading="lazy"> The convergence of the integral on this interval can be proven by Dirichlet's test after substitution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle u={\tfrac {t^{3}}{3}}+xt.}">

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<annotation encoding="application/x-tex">{\textstyle u={\tfrac {t^{3}}{3}}+xt.}</annotation>

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y = Ai(x) satisfies the Airy equation

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This equation has two <a href="Linear_independence" title="Linear independence">linearly independent</a> solutions.

Up to <a href="Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a>, 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 the same amplitude of oscillation as Ai(x) as x → −∞ which differs in phase by π/2:

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<annotation encoding="application/x-tex">{\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.}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

The values of Ai(x) and Bi(x) and their derivatives at x = 0 are given by

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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Here, Γ denotes the <a href="Gamma_function" title="Gamma function">Gamma function</a>. It follows that the <a href="Wronskian" title="Wronskian">Wronskian</a> of Ai(x) and Bi(x) is 1/π.

When x is positive, Ai(x) is positive, <a href="Convex_function" title="Convex function">convex</a>, 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 the asymptotic formulae below for the Airy functions.

The Airy functions are orthogonal<a href="#cite_note-1">[1]</a> in the sense that

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<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }\operatorname {Ai} (t+x)\operatorname {Ai} (t+y)dt=\delta (x-y)}</annotation>

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again using an improper Riemann integral.

<dl><dt>Real zeros of Ai(x) and its derivative Ai'(x)</dt></dl>

Neither Ai(x) nor its <a href="Derivative" title="Derivative">derivative</a> Ai'(x) have positive real zeros. The "first" real zeros (i.e. nearest to x=0) are:<a href="#cite_note-2">[2]</a>

<ul><li>"first" zeros of Ai(x) are at x ≈ −2.33811, −4.08795, −5.52056, −6.78671, ...</li>

<li>"first" zeros of its derivative Ai'(x) are at x ≈ −1.01879, −3.24820, −4.82010, −6.16331, ...</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Asymptotic_formulae">Asymptotic formulae</h2></div>

As explained below, the Airy functions can be extended to the <a href="Complex_plane" title="Complex plane">complex plane</a>, giving <a href="Entire_function" title="Entire function">entire functions</a>. The asymptotic behaviour of the Airy functions as |z| goes to infinity at a constant value of <a href="Arg_(mathematics)" class="mw-redirect" title="Arg (mathematics)">arg</a>(z) depends on arg(z): this is called the <a href="Stokes_phenomenon" title="Stokes phenomenon">Stokes phenomenon</a>. For |arg(z)| < π we have the following <a href="Asymptotic_formula" class="mw-redirect" title="Asymptotic formula">asymptotic formula</a> for Ai(z):<a href="#cite_note-:0-3">[3]</a>

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<annotation encoding="application/x-tex">{\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].}</annotation>

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or<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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].}">

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<annotation encoding="application/x-tex">{\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].}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta ={\tfrac {2}{3}}z^{3/2}.}">

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<annotation encoding="application/x-tex">{\displaystyle \zeta ={\tfrac {2}{3}}z^{3/2}.}</annotation>

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</math><img src="./e34d48b0330630d8fe4ca824592a61ee58447a8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.287ex; height:3.843ex;" alt="{\displaystyle \zeta ={\tfrac {2}{3}}z^{3/2}.}" loading="lazy"> In particular, the first few terms are<a href="#cite_note-4">[4]</a><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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)}">

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<annotation encoding="application/x-tex">{\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)}</annotation>

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There is a similar one for Bi(z), but only applicable when |arg(z)| < π/3:

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<annotation encoding="application/x-tex">{\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].}</annotation>

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A more accurate formula for Ai(z) and a formula for Bi(z) when π/3 < |arg(z)| < π or, equivalently, for Ai(−z) and Bi(−z) when |arg(z)| < 2π/3 but not zero, are:<a href="#cite_note-:0-3">[3]</a><a href="#cite_note-:1-5">[5]</a><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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}}}">

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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When |arg(z)| = 0 these are good approximations but are not asymptotic because the ratio between Ai(−z) or Bi(−z) and the above approximation goes to infinity whenever the sine or cosine goes to zero.

<a href="Asymptotic_analysis" title="Asymptotic analysis">Asymptotic expansions</a> 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:<a href="#cite_note-:1-5">[5]</a>

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<annotation encoding="application/x-tex">{\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].}</annotation>

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When |arg(z)| < π/3 we have:<a href="#cite_note-:1-5">[5]</a>

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<annotation encoding="application/x-tex">{\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].}</annotation>

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Similarly, an expression for Ai'(−z) and Bi'(−z) when |arg(z)| < 2π/3 but not zero, are<a href="#cite_note-:1-5">[5]</a>

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Complex_arguments">Complex arguments</h2></div>

We can extend the definition of the Airy function to the complex plane by

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Ai} (z)={\frac {1}{2\pi i}}\int _{C}\exp \left({\tfrac {t^{3}}{3}}-zt\right)\,dt,}</annotation>

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where 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 <a href="Entire_function" title="Entire function">entire functions</a> on the complex plane.

The asymptotic formula for Ai(x) is still valid in the complex plane if the principal value of x2/3 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {C} :\left|\arg(x)\right|<{\tfrac {\pi }{3}}-\delta }">

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<annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {C} :\left|\arg(x)\right|<{\tfrac {\pi }{3}}-\delta }</annotation>

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</math><img src="./8b5ceae4a7cc59ec822b980876ecc25e24e86b4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:24.22ex; height:3.343ex;" alt="{\displaystyle x\in \mathbb {C} :\left|\arg(x)\right|<{\tfrac {\pi }{3}}-\delta }" loading="lazy"> for some positive δ. Finally, the formulae for Ai(−x) and Bi(−x) are valid if x is in the sector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {C} :\left|\arg(x)\right|<{\tfrac {2\pi }{3}}-\delta .}">

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<annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {C} :\left|\arg(x)\right|<{\tfrac {2\pi }{3}}-\delta .}</annotation>

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It follows from the asymptotic behaviour of the Airy functions that both Ai(x) and Bi(x) have an infinity of zeros on 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \mathbb {C} :{\tfrac {\pi }{3}}<\left|\arg(z)\right|<{\tfrac {\pi }{2}}.}">

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<annotation encoding="application/x-tex">{\displaystyle z\in \mathbb {C} :{\tfrac {\pi }{3}}<\left|\arg(z)\right|<{\tfrac {\pi }{2}}.}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Plots">Plots</h3></div>

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<annotation encoding="application/x-tex">{\displaystyle \Re \left[\operatorname {Ai} (x+iy)\right]}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \Im \left[\operatorname {Ai} (x+iy)\right]}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \left|\operatorname {Ai} (x+iy)\right|\,}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {arg} \left[\operatorname {Ai} (x+iy)\right]\,}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \Re \left[\operatorname {Bi} (x+iy)\right]}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \Im \left[\operatorname {Bi} (x+iy)\right]}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \left|\operatorname {Bi} (x+iy)\right|\,}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {arg} \left[\operatorname {Bi} (x+iy)\right]\,}</annotation>

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</math><img src="./ef422f3f85b66c485cf5b87ffa8b57b1b3e254a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.147ex; height:2.843ex;" alt="{\displaystyle \operatorname {arg} \left[\operatorname {Bi} (x+iy)\right]\,}" loading="lazy">

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<div class="mw-heading mw-heading2"><h2 id="Relation_to_other_special_functions">Relation to other special functions</h2></div>

For positive arguments, the Airy functions are related to the <a href="Bessel_function#Modified_Bessel_functions" title="Bessel function">modified Bessel functions</a>:

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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Here, I±1/3 and K1/3 are solutions of <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}y''+xy'-\left(x^{2}+{\tfrac {1}{9}}\right)y=0.}">

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<annotation encoding="application/x-tex">{\displaystyle x^{2}y''+xy'-\left(x^{2}+{\tfrac {1}{9}}\right)y=0.}</annotation>

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The first derivative of the Airy function is

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Ai'} (x)=-{\frac {x}{\pi {\sqrt {3}}}}\,K_{2/3}\!\left({\frac {2}{3}}x^{3/2}\right).}</annotation>

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Functions K1/3 and K2/3 can be represented in terms of rapidly convergent integrals<a href="#cite_note-6">[6]</a> (see also <a href="Bessel_function#Modified_Bessel_functions" title="Bessel function">modified Bessel functions</a>)

For negative arguments, the Airy function are related to the <a href="Bessel_function" title="Bessel function">Bessel functions</a>:

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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Here, J±1/3 are solutions of

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<annotation encoding="application/x-tex">{\displaystyle x^{2}y''+xy'+\left(x^{2}-{\frac {1}{9}}\right)y=0.}</annotation>

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The <a href="Scorer's_function" title="Scorer's function">Scorer's functions</a> Hi(x) and -Gi(x) solve the equation y′′ − xy = 1/π. They can also be expressed in terms of the Airy functions:

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Fourier_transform">Fourier transform</h2></div>

Using the definition of the Airy function Ai(x), it is straightforward to show that its <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> is given by

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<annotation encoding="application/x-tex">{\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}}.}</annotation>

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</math>This can be obtained by taking the Fourier transform of the Airy equation. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\hat {y}}={\frac {1}{2\pi i}}\int ye^{-ikx}dx}">

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<annotation encoding="application/x-tex">{\textstyle {\hat {y}}={\frac {1}{2\pi i}}\int ye^{-ikx}dx}</annotation>

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</math><img src="./0404bc2461504285acd6eac2f476feeca7a36164.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.42ex; height:3.509ex;" alt="{\textstyle {\hat {y}}={\frac {1}{2\pi i}}\int ye^{-ikx}dx}" loading="lazy">. Then, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i{\hat {y}}'+k^{2}{\hat {y}}=0}">

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<annotation encoding="application/x-tex">{\displaystyle i{\hat {y}}'+k^{2}{\hat {y}}=0}</annotation>

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</math><img src="./ecb54b48514924b56d7aaf2df5e9b067a6aa4e61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.458ex; height:3.009ex;" alt="{\displaystyle i{\hat {y}}'+k^{2}{\hat {y}}=0}" loading="lazy">, which then has solutions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {y}}=Ce^{ik^{3}/3}.}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {y}}=Ce^{ik^{3}/3}.}</annotation>

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</math><img src="./c73c1f7829262828920ab050ebe41696f6ef57e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.029ex; height:3.343ex;" alt="{\displaystyle {\hat {y}}=Ce^{ik^{3}/3}.}" loading="lazy"> There is only one dimension of solutions because the Fourier transform requires y to decay to zero fast enough; Bi grows to infinity exponentially fast, so it cannot be obtained via a Fourier transform.

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Quantum_mechanics">Quantum mechanics</h3></div>

The Airy function is the solution to the <a href="Time-independent_Schr%C3%B6dinger_equation" class="mw-redirect" title="Time-independent Schrödinger equation">time-independent Schrödinger equation</a> for a particle confined within a triangular <a href="Potential_well" title="Potential well">potential well</a> and for a particle in a one-dimensional constant force field. For the same reason, it also serves to provide uniform semiclassical approximations near a turning point in the <a href="WKB_approximation" title="WKB approximation">WKB approximation</a>, 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 <a href="Heterojunction" title="Heterojunction">heterojunctions</a>.

<div class="mw-heading mw-heading3"><h3 id="Optics">Optics</h3></div>

A transversally asymmetric optical beam, where the <a href="Electric_field" title="Electric field">electric field</a> profile is given by the Airy function, has the interesting property that its maximum intensity accelerates towards one side instead of propagating in 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.

<div class="mw-heading mw-heading3"><h3 id="Caustics">Caustics</h3></div>

The Airy function underlies the form of the intensity near an optical directional <a href="Caustic_(optics)" title="Caustic (optics)">caustic</a>, such as that of the <a href="Rainbow" title="Rainbow">rainbow</a> (called supernumerary rainbow). Historically, this was the mathematical problem that led Airy to develop this special function. In 1841, <a href="William_Hallowes_Miller" title="William Hallowes Miller">William Hallowes Miller</a> experimentally measured 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.<a href="#cite_note-7">[7]</a>

<div class="mw-heading mw-heading3"><h3 id="Probability">Probability</h3></div>

In the mid-1980s, the Airy function was found to be intimately connected to <a href="Chernoff's_distribution" title="Chernoff's distribution">Chernoff's distribution</a>.<a href="#cite_note-8">[8]</a>

The Airy function also appears in the definition of <a href="Tracy%E2%80%93Widom_distribution" title="Tracy–Widom distribution">Tracy–Widom distribution</a> which describes the law of largest eigenvalues in <a href="Random_matrix" title="Random matrix">Random matrix</a>. Due to the intimate connection of random matrix theory with the <a href="Kardar%E2%80%93Parisi%E2%80%93Zhang_equation" title="Kardar–Parisi–Zhang equation">Kardar–Parisi–Zhang equation</a>, there are central processes constructed in KPZ such as the <a href="Airy_process" title="Airy process">Airy process</a>.<a href="#cite_note-9">[9]</a>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

The Airy function is named after the British astronomer and physicist <a href="George_Biddell_Airy" title="George Biddell Airy">George Biddell Airy</a> (1801–1892), who encountered it in his early study of <a href="Optics" title="Optics">optics</a> in physics (Airy 1838). The notation Ai(x) was introduced by <a href="Harold_Jeffreys" title="Harold Jeffreys">Harold Jeffreys</a>. Airy had become the British <a href="Astronomer_Royal" title="Astronomer Royal">Astronomer Royal</a> in 1835, and he held that post until his retirement in 1881.

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<ul><li><a href="Airy_zeta_function" title="Airy zeta function">Airy zeta function</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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</style><cite id="CITEREFAspnes1966" class="citation journal cs1">Aspnes, David E. (1966). "Electric-Field Effects on Optical Absorption near Thresholds in Solids". Physical Review. 147 (2): 554–566. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.147.554">10.1103/PhysRev.147.554</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-899X">0031-899X</a>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://dlmf.nist.gov/9.9">"Airy and Related Function"</a>. dlmf.nist.gov. Retrieved 9 October 2022.</cite>

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<li id="cite_note-:0-3">^ <a href="#cite_ref-:0_3-0">a</a> <a href="#cite_ref-:0_3-1">b</a> <a href="#CITEREFAbramowitzStegun1983">Abramowitz & Stegun (1983</a>, p. 448), Eqns 10.4.59, 10.4.61

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://dlmf.nist.gov/9.7">"DLMF: §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions"</a>. dlmf.nist.gov. Retrieved 2023-05-11.</cite>

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<li id="cite_note-:1-5">^ <a href="#cite_ref-:1_5-0">a</a> <a href="#cite_ref-:1_5-1">b</a> <a href="#cite_ref-:1_5-2">c</a> <a href="#cite_ref-:1_5-3">d</a> <a href="#CITEREFAbramowitzStegun1983">Abramowitz & Stegun (1983</a>, p. 448), Eqns 10.4.60 and 10.4.64

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> M.Kh.Khokonov. Cascade Processes of Energy Loss by Emission of Hard Photons // JETP, V.99, No.4, pp. 690-707 \ (2004).

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<li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="https://archive.org/details/transactionsofca07camb/page/n249/mode/2up" class="extiw external" title="iarchive:transactionsofca07camb/page/n249/mode/2up">Miller, William Hallowes. "On spurious rainbows." Transactions of the Cambridge Philosophical Society 7 (1848): 277.</a>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFGroeneboomLalleyTemme2015" class="citation journal cs1">Groeneboom, Piet; Lalley, Steven; Temme, Nico (2015). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jmaa.2014.10.051">"Chernoff's distribution and differential equations of parabolic and Airy type"</a>. <a href="Journal_of_Mathematical_Analysis_and_Applications" title="Journal of Mathematical Analysis and Applications">Journal of Mathematical Analysis and Applications</a>. 423 (2): 1804–1824. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1305.6053">1305.6053</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jmaa.2014.10.051">10.1016/j.jmaa.2014.10.051</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119173815">119173815</a>.</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFQuastelRemenik2014" class="citation book cs1">Quastel, Jeremy; Remenik, Daniel (2014). <a rel="nofollow" class="external text" href="https://link.springer.com/chapter/10.1007/978-1-4939-0339-9_5">"Airy Processes and Variational Problems"</a>. Topics in Percolative and Disordered Systems. Springer Proceedings in Mathematics & Statistics. Vol. 69. pp. 121–171. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1301.0750">1301.0750</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4939-0339-9_5">10.1007/978-1-4939-0339-9_5</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4939-0338-2</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118241762">118241762</a>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFAbramowitzStegun1983" class="citation book cs1"><a href="Milton_Abramowitz" title="Milton Abramowitz">Abramowitz, Milton</a>; <a href="Irene_Stegun" title="Irene Stegun">Stegun, Irene Ann</a>, eds. (1983) [June 1964]. <a rel="nofollow" class="external text" href="http://www.math.ubc.ca/~cbm/aands/page_448.htm">"Chapter 10"</a>. <a href="Abramowitz_and_Stegun" title="Abramowitz and Stegun">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a>. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 448. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-61272-0</bdi>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/64-60036">64-60036</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0167642">0167642</a>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://www.loc.gov/item/65012253">65-12253</a>.</cite></li>

<li><cite id="CITEREFAiry1838" class="citation cs2">Airy (1838), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-yI8AAAAMAAJ&q=Transactions+of+the+Cambridge+Philosophical+Society+1838">"On the intensity of light in the neighbourhood of a caustic"</a>, Transactions of the Cambridge Philosophical Society, 6, University Press: 379–402, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1838TCaPS...6..379A">1838TCaPS...6..379A</a></cite></li>

<li><a href="Frank_William_John_Olver" class="mw-redirect" title="Frank William John Olver">Frank William John Olver</a> (1974). Asymptotics and Special Functions, Chapter 11. Academic Press, New York.</li>

<li><cite id="CITEREFPressTeukolskyVetterlingFlannery2007" class="citation cs2">Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110811154417/http://apps.nrbook.com/empanel/index.html#pg=289">"Section 6.6.3. Airy Functions"</a>, Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-88068-8</bdi>, archived from <a rel="nofollow" class="external text" href="http://apps.nrbook.com/empanel/index.html#pg=289">the original</a> on 2011-08-11, retrieved 2011-08-09</cite></li>

<li><cite id="CITEREFValléeSoares2004" class="citation cs2">Vallée, Olivier; Soares, Manuel (2004), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100113044654/http://worldscibooks.com/physics/p345.html">Airy functions and applications to physics</a>, London: Imperial College Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-86094-478-9</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2114198">2114198</a>, archived from <a rel="nofollow" class="external text" href="http://www.worldscibooks.com/physics/p345.html">the original</a> on 2010-01-13, retrieved 2010-05-14</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Airy_functions">"Airy functions"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>

<li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/AiryFunctions.html">"Airy Functions"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li>

<li>Wolfram function pages for <a rel="nofollow" class="external text" href="http://functions.wolfram.com/Bessel-TypeFunctions/AiryAi/">Ai</a> and <a rel="nofollow" class="external text" href="http://functions.wolfram.com/Bessel-TypeFunctions/AiryBi/">Bi</a> functions. Includes formulas, function evaluator, and plotting calculator.</li>

<li><cite id="CITEREFOlver2010" class="citation cs2">Olver, F. W. J. (2010), <a rel="nofollow" class="external text" href="http://dlmf.nist.gov/9">"Airy and related functions"</a>, in <a href="Frank_W._J._Olver" title="Frank W. J. Olver">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <a href="Digital_Library_of_Mathematical_Functions" title="Digital Library of Mathematical Functions">NIST Handbook of Mathematical Functions</a>, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-19225-5</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a></cite>.</li></ul>

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