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# airy

Airy function

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

## Input Arguments

 x Symbolic number, variable, expression, or vector or matrix of symbolic numbers, variables, expressions.

## 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.

## 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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