airy

Airy Functions

Syntax

W = airy(Z)

W = airy(k,Z)

W = airy(k,Z,scale)

Description

W = airy(Z) returns the Airy function, Ai(Z), for each element of Z.

W = airy(k,Z) returns any of four different Airy functions, depending on the value of k, such as the Airy function of the second kind or the first derivative of an Airy function.

W = airy(k,Z,scale) scales the resulting Airy function. airy applies a specific scaling function to W depending on your choice of k and scale.

Examples

Airy Function of Real-Valued x

Define x.

x = -10:0.01:1;

Calculate Ai(x)

ai = airy(x);

Calculate Bi(x) using $$k=2$$.

bi = airy(2,x);

Plot both results together on the same axes.

figure
plot(x,ai,'-b',x,bi,'-r')
axis([-10 1 -0.6 1.4])
xlabel('x')
legend('Ai(x)','Bi(x)','Location','NorthWest')

Airy Function of Complex-Valued x

Compute the Airy function at a slice through the complex plane at $$x+i$$.

Take a slice through the complex plane.

x = -4:0.1:4;
z = x+1i;

Calculate Ai(z).

w = airy(z);

Plot the real part of the result.

figure
plot(x, real(w))
axis([-4 4 -1.5 1])
xlabel('real(z)')

Scaled Airy Function

Define x.

x = -10:0.01:1;

Calculate the scaled and unscaled Airy function.

scaledAi = airy(0,x,1);
noscaleAi =  airy(0,x,0);

Plot the real part of each result.

rscaled = real(scaledAi);
rnoscale = real(noscaleAi);
figure
plot(x,rscaled,'-b',x,rnoscale,'-r')
axis([-10 1 -0.60 0.60])
xlabel('x')
legend('scaled','not scaled','Location','SouthEast')

Input Arguments

Z — System variable

vector | matrix | N-D Array

System variable, specified as a real or complex vector, matrix, or N-D array.

Data Types: single | double
Complex Number Support: Yes

k — Type of Airy function

0 (default) | 1 | 2 | 3

Type of Airy function, specified as one of four values.

k	Returns
0	Airy function,$$Ai(Z)$$, which is the same as airy(Z).
1	First derivative of Airy function, $$A{i}^{\prime }(Z)$$.
2	Airy function of the second kind,$$Bi(Z)$$
3	First derivative of Airy function of the second kind,$$B{i}^{\prime }(Z)$$

Data Types: single | double

scale — Scaling option

0 (default) | 1

Scaling option, specified as 0 or 1. Use scale = 1 to enable the scaling of Z. The values you specify for k and scale determine the scaling function airy applies to Z.

scale	k	Scaling applied to output
0	Any	None
1	0 or 1	$$e^{\frac{2}{3}{Z}^{(3/2)}}$$
1	2 or 3	$$e^{-\left|\frac{2}{3}Re(Z^{(3/2)})\right|}$$

Data Types: single | double

Output Arguments

W — Airy function of Z

vector | matrix | N-D Array

Airy function of Z, returned as an array the same size as Z.

More About

Airy Functions

The Airy functions form a pair of linearly independent solutions to

$$\frac{d^{2}W}{d{Z}^2}-ZW=0.$$

The relationship between the Airy and modified Bessel functions is

$$\begin{array}{l}Ai(Z)=\left[\frac{1}{\pi }\sqrt{\frac{Z}{3}}\right]K_{1/3}(\zeta )\\ Bi(Z)=\sqrt{\frac{Z}{3}}\left[I_{-1/3}(\zeta )+I_{1/3}(\zeta )\right],\end{array}$$

where

$$\zeta =\frac{2}{3}{Z}^{3/2}.$$

References

[1] Amos, D. E. “Algorithm 644: A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order.” ACM Transactions on Mathematical Software 12, no. 3 (September 1986): 265–273. https://dl.acm.org/doi/10.1145/7921.214331.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• Always returns a complex result.

• Strict single-precision calculations are not supported. In the generated code, single-precision inputs produce single-precision outputs. However, variables inside the function might be double-precision.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

Version History

Introduced before R2006a