airy

Airy function

Syntax

airy(x)

airy(0,x)

airy(1,x)

airy(2,x)

airy(3,x)

airy(n,x)

airy(___,1)

Description

airy(x) returns the Airy function of the first kind, Ai(x), for each element of x.

airy(0,x) is the same as airy(x).

airy(1,x) returns the derivative of Ai(x).

airy(2,x) returns the Airy function of the second kind, Bi(x).

airy(3,x) returns the derivative of Bi(x).

airy(n,x) uses the values in vector n to return the corresponding Airy functions of elements of vector x. Both n and x must have the same size.

airy(___,1) returns the Scaled Airy Functions following the syntax for the MATLAB^® airy function.

Examples

Find the Airy Function of the First Kind

Find the Airy function of the first kind, Ai(x), for numeric or symbolic inputs using airy. Approximate exact symbolic outputs using vpa.

Find the Airy function of the first kind, Ai(x), at 1.5. Because the input is double and not symbolic, you get a double result.

airy(1.5)

ans =
    0.0717

Find the Airy function of the values of vector v symbolically, by converting v to symbolic form using sym. Because the input is symbolic, airy returns exact symbolic results. The exact symbolic results for most symbolic inputs are unresolved function calls.

v = sym([-1 0 25.1 1+1i]);
vAiry = airy(v)

vAiry =
[ airy(0, -1), 3^(1/3)/(3*gamma(2/3)), airy(0, 251/10), airy(0, 1 + 1i)]

Numerically approximate the exact symbolic result using vpa.

vpa(vAiry)

ans =
[ 0.53556088329235211879951656563887, 0.35502805388781723926006318600418,...
 4.9152763177499054787371976959487e-38,...
 0.060458308371838149196532978116646 - 0.15188956587718140235494791259223i]

Find the Airy function, Ai(x), of the symbolic input x^2. For symbolic expressions, airy returns an unresolved call.

syms x
airy(x^2)

ans =
airy(0, x^2)

Find the Airy Function of the Second Kind

Find the Airy function of the second kind, Bi(x), of the symbolic input [-3 4 1+1i x^2] by specifying the first argument as 2. Because the input is symbolic, airy returns exact symbolic results. The exact symbolic results for most symbolic inputs are unresolved function calls.

v = sym([-3 4 1+1i x^2]);
vAiry = airy(2, v)

vAiry =
[ airy(2, -3), airy(2, 4), airy(2, 1 + 1i), airy(2, x^2)]

Use the syntax airy(2,x) like airy(x), as described in the example Find the Airy Function of the First Kind.

Plot Airy Functions

Plot the Airy Functions, $$Ai(x)$$ and $$Bi(x)$$, over the interval [-10 2] using fplot.

syms x
fplot(airy(x), [-10 2])
hold on
fplot(airy(2,x), [-10 2])
legend('Ai(x)','Bi(x)','Location','Best')
title('Airy functions Ai(x) and Bi(x)')
grid on
hold off

Plot the absolute value of $$Ai(z)$$ over the complex plane.

syms y
z = x + 1i*y;
fsurf(abs(airy(z)),[-3 3 -3 3])
title('|Ai(z)|')

Find Derivatives of Airy Functions

Find the derivative of the Airy function of the first kind, Ai′(x), at 0 by specifying the first argument of airy as 1. Then, numerically approximate the derivative using vpa.

dAi = airy(1, sym(0))
dAi_vpa = vpa(dAi)

dAi =
-(3^(1/6)*gamma(2/3))/(2*pi)
dAi_vpa =
-0.2588194037928067984051835601892

Find the derivative of the Airy function of the second kind, Bi′(x), at x by specifying the first argument as 3. Then, find the derivative at x = 5 by substituting for x using subs and calling vpa.

syms x
dBi = airy(3, x)
dBi_vpa = vpa(subs(dBi, x, 5))

dBi =
airy(3, x)
dBi_vpa =
1435.8190802179825186717212380046

Solve Airy Differential Equation for Airy Functions

Show that the Airy functions Ai(x) and Bi(x) are the solutions of the differential equation

$$\frac{{\partial }^{2}y}{\partial x^2}-xy=0.$$

syms y(x)
dsolve(diff(y, 2) - x*y == 0)

ans =
C1*airy(0, x) + C2*airy(2, x)

Differentiate Airy Functions

Differentiate expressions containing airy.

syms x y
diff(airy(x^2))
diff(diff(airy(3, x^2 + x*y -y^2), x), y)

ans =
2*x*airy(1, x^2)
 
ans =
airy(2, x^2 + x*y - y^2)*(x^2 + x*y - y^2) +...
airy(2, x^2 + x*y - y^2)*(x - 2*y)*(2*x + y) +...
airy(3, x^2 + x*y - y^2)*(x - 2*y)*(2*x + y)*(x^2 + x*y - y^2)
 

Expand Airy Function using Taylor Series

Find the Taylor series expansion of the Airy functions, Ai(x) and Bi(x), using taylor.

aiTaylor = taylor(airy(x))
biTaylor = taylor(airy(2, x))

aiTaylor =
- (3^(1/6)*gamma(2/3)*x^4)/(24*pi) + (3^(1/3)*x^3)/(18*gamma(2/3))...
 - (3^(1/6)*gamma(2/3)*x)/(2*pi) + 3^(1/3)/(3*gamma(2/3))
biTaylor =
(3^(2/3)*gamma(2/3)*x^4)/(24*pi) + (3^(5/6)*x^3)/(18*gamma(2/3))...
 + (3^(2/3)*gamma(2/3)*x)/(2*pi) + 3^(5/6)/(3*gamma(2/3))

Fourier Transform of Airy Function

Find the Fourier transform of the Airy function Ai(x) using fourier.

syms x
aiFourier = fourier(airy(x))

aiFourier =
exp((w^3*1i)/3)

Numeric Roots of Airy Function

Find a root of the Airy function Ai(x) numerically using vpasolve.

syms x
vpasolve(airy(x) == 0, x)

ans =
 -226.99630507523600716771890962744

Find a root in the interval [-5 -3].

vpasolve(airy(x) == 0, x, [-5 -3])

ans =
-4.0879494441309706166369887014574

Input Arguments

x — Input

number | vector | matrix | multidimensional array | symbolic number | symbolic variable | symbolic vector | symbolic matrix | symbolic multidimensional array | symbolic function | symbolic expression

Input, specified as a number, vector, matrix, or multidimensional array, or a symbolic number, variable, vector, matrix, multidimensional array, function, or expression.

n — Type of Airy function

0 (default) | number | vector | matrix | multidimensional array | symbolic number | symbolic variable | symbolic vector | symbolic matrix | symbolic multidimensional array

Type of Airy function, specified as a number, vector, matrix, or multidimensional array, or a symbolic number, variable, vector, matrix, or multidimensional array. The values of the input must be 0, 1, 2, or 3, which specify the Airy function as follows.

n	Returns
0 (default)	Airy function, Ai(x), which is the same as airy(x).
1	Derivative of Airy function, Ai’(x).
2	Airy function of the second kind, Bi(x).
3	Derivative of Airy function of the second kind, Bi’(x).

More About

Airy Functions

The Airy functions Ai(x) and Bi(x) are the two linearly independent solutions of the differential equation

$$\frac{{\partial }^{2}y}{\partial x^2}-xy=0.$$

Ai(x) is called the Airy function of the first kind. Bi(x) is called the Airy function of the second kind.

Scaled Airy Functions

The Airy function of the first kind, Ai(x), is scaled as

$$e^{\left(\frac{2}{3}{x}^{\left(3/2\right)}\right)}\text{Ai}\left(x\right).$$

The derivative, Ai’(x), is scaled by the same factor.

The Airy function of the second kind, Bi(x), is scaled as

$$e^{-\left|\frac{2}{3}\text{Re}\left(x^{\left(3/2\right)}\right)\right|}\text{Bi}\left(x\right).$$

The derivative, Bi’(x), is scaled by the same factor.

Tips

• When you call airy for inputs that are not symbolic objects, you call the MATLAB airy function.

• When you call airy(n, x), at least one argument must be a scalar or both arguments must be vectors or matrices of the same size. If one argument is a scalar and the other is a vector or matrix, airy(n,x) expands the scalar into a vector or matrix of the same size as the other argument with all elements equal to the scalar.

• airy returns special exact values at 0.

Version History

Introduced in R2012a