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airy

Airy function

Syntax

airy(x)
airy(0,x)
airy(1,x)
airy(2,x)
airy(3,x)
airy(n,x)

Description

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

airy(0,x) is equivalent to airy(x).

airy(1,x) returns the derivative of the Airy function of the first kind, Ai′(x).

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

airy(3,x) returns the derivative of the Airy function of the second kind, Bi′(x).

airy(n,x) returns a vector or matrix of derivatives of the Airy function.

Input Arguments

x

Symbolic number, variable, expression, or function, or a vector or matrix of symbolic numbers, variables, expressions, or functions. If x is a vector or matrix, airy returns the Airy functions for each element of x.

n

Vector or matrix of numbers 0, 1, 2, and 3.

Examples

Solve this second-order differential equation. The solutions are the Airy functions of the first and the second kind.

syms y(x)
dsolve(diff(y, 2) - x*y == 0)
ans =
C2*airy(0, x) + C3*airy(2, x)
 

Verify that the Airy function of the first kind is a valid solution of the Airy differential equation:

syms x
simplify(diff(airy(0, x), x, 2) - x*airy(0, x)) == 0
ans =
     1

Verify that the Airy function of the second kind is a valid solution of the Airy differential equation:

simplify(diff(airy(2, x), x, 2) - x*airy(2, x)) == 0
ans =
     1
 

Compute the Airy functions for these numbers. Because these numbers are not symbolic objects, you get floating-point results.

[airy(1), airy(1, 3/2 + 2*i), airy(2, 2), airy(3, 1/101)]
ans =
   0.1353             0.1641 + 0.1523i   3.2981             0.4483
 

Compute the Airy functions for the numbers converted to symbolic objects. For most symbolic (exact) numbers, airy returns unresolved symbolic calls.

[airy(sym(1)), airy(1, sym(3/2 + 2*i)), airy(2, sym(2)), airy(3, sym(1/101))]
ans =
[ airy(0, 1), airy(1, 3/2 + 2*i), airy(2, 2), airy(3, 1/101)]

For symbolic variables and expressions, airy also returns unresolved symbolic calls:

syms x y
[airy(x), airy(1, x^2), airy(2, x - y), airy(3, x*y)]
ans =
[ airy(0, x), airy(1, x^2), airy(2, x - y), airy(3, x*y)]
 

Compute the Airy functions for x = 0. The Airy functions have special values for this parameter.

airy(sym(0))
ans =
3^(1/3)/(3*gamma(2/3))
airy(1, sym(0))
ans =
-(3^(1/6)*gamma(2/3))/(2*pi)
airy(2, sym(0))
ans =
3^(5/6)/(3*gamma(2/3))
airy(3, sym(0))
ans =
(3^(2/3)*gamma(2/3))/(2*pi)

If you do not use sym, you call the MATLAB® airy function that returns numeric approximations of these values:

[airy(0), airy(1, 0), airy(2, 0), airy(3, 0)]
ans =
    0.3550   -0.2588    0.6149    0.4483
 

Differentiate the expressions involving the Airy functions:

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)
 
 

Compute the Airy function of the first kind for the elements of matrix A:

syms x
A = [-1, 0; 0, x];
airy(A)
ans =
[            airy(0, -1), 3^(1/3)/(3*gamma(2/3))]
[ 3^(1/3)/(3*gamma(2/3)),             airy(0, x)]
 

Plot the Airy function Ai(x) and its derivative Ai'(x):

syms x
ezplot(airy(x));
hold on

p = ezplot(airy(1,x));
set(p,'Color','red')

title('Airy function Ai and its first derivative')
hold off

More About

expand all

Airy Functions

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

Tips

  • Calling airy for a number that is not a symbolic object invokes the MATLAB airy function.

  • When you call airy with two input arguments, at least one argument must be a scalar or both arguments must be vectors or matrices of the same size. If one input argument is a scalar and the other one is a vector or a 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 that scalar.

References

Antosiewicz, H. A. "Bessel Functions of Fractional Order." Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. (M. Abramowitz and I. A. Stegun, eds.). New York: Dover, 1972.

See Also

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