airy

Syntax

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

Description

example

````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)`.```

example

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

example

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

example

````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, $Ai\left(x\right)$ and $Bi\left(x\right)$, 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```

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

```syms y z = x + 1i*y; figure(2) fsurf(abs(airy(z))) title('|Ai(z)|') a = gca; a.ZLim = [0 10]; caxis([0 10])```

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

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

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).

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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

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

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`.