airy — Airy | CodeCogs

Calculates the Airy functions Ai and Bi and their derivatives.

Interface

#include <codecogs/maths/special/airy/airy.h>

using namespace Maths::Special::Airy;

int airy(double x, double &Ai, double &Aip, double &Bi, double &Bip )

There are four varieties of Airy functions: Ai(x), Bi(x), Gi(x), and Hi(x). Of these, Ai(x) and Bi(x) are by far the most common, with Gi(x) and Hi(x) being encountered much less frequently. Airy functions commonly appear in physics, especially in optics, quantum mechanics, electromagnetics, and radiative transfer.

Figure 1: Plots of the two most commonly used Airy functions

Ai(x) and Bi(x) are defined as the two linearly independent solutions to

$$\frac{{\rm d}^2 y}{\rm d y^2} - x y = 0$$

(1)

in the form

$$y(x) = A Ai(x) + B Bi(x)$$

(2)

where

$$Ai(x) = \frac{1}{3} \sqrt{x} \left [ I_{-1/3} \left ( \frac{2}{3} x^{3/2} \right ) - I_{1/3} \left ( \frac{2}{3} x^{3/2} \right ) \right ]$$

(3)

$$Bi(x) = \sqrt{\frac{x}{3}} \left [ I_{-1/3} \left ( \frac{2}{3} x^{3/2} \right ) + I_{1/3} \left ( \frac{2}{3} x^{3/2} \right ) \right ]$$

(4)

The function returns the two independent solutions Ai(x), Bi(x) and their first derivatives Ai'(x), Bi'(x).

Evaluation is by power series summation for small x, by rational minimax approximations for large x.

Accuracy:

Error criterion is absolute when function <= 1, relative when function > 1, except * denotes relative error criterion. For large negative x, the absolute error increases as x^1.5. For large positive x, the relative error increases as x^1.5.

<pre> Arithmetic domain function # trials peak rms IEEE -10, 0 Ai 10000 1.6e-15 2.7e-16 IEEE 0, 10 Ai 10000 2.3e-14* 1.8e-15* IEEE -10, 0 Ai' 10000 4.6e-15 7.6e-16 IEEE 0, 10 Ai' 10000 1.8e-14* 1.5e-15* IEEE -10, 10 Bi 30000 4.2e-15 5.3e-16 IEEE -10, 10 Bi' 30000 4.9e-15 7.3e-16 </pre>

References

Cephes Math Library Release 2.8: June, 2000; http://mathworld.wolfram.com/AiryFunctions.html

Example 1

#include <stdio.h>
#include <codecogs/maths/special/airy/airy.h>

int main()
{
  double Ai, Bi, Aip, Bip;
  for(double x=-10; x<2;x++)
  {
    Maths::Special::Airy::airy(x, Ai, Aip, Bi, Bip);
    printf("\n Airy(%.0lf) -> Ai=%lf Bi=%lf Ai'=%lf Bi'=%lf",x,Ai, Bi, Aip, Bip);
  }
  return 0;
}

Output:

Airy(-10) -> Ai=0.040241 Bi=-0.314680 Ai'=0.996265 Bi'=0.119414
Airy(-9) -> Ai=-0.022134 Bi=0.324947 Ai'=-0.975664 Bi'=-0.057401
Airy(-8) -> Ai=-0.052705 Bi=-0.331252 Ai'=0.935561 Bi'=-0.159450
Airy(-7) -> Ai=0.184281 Bi=0.293762 Ai'=-0.771008 Bi'=0.498245
Airy(-6) -> Ai=-0.329145 Bi=-0.146698 Ai'=0.345935 Bi'=-0.812899
Airy(-5) -> Ai=0.350761 Bi=-0.138369 Ai'=0.327193 Bi'=0.778412
Airy(-4) -> Ai=-0.070266 Bi=0.392235 Ai'=-0.790629 Bi'=-0.116671
Airy(-3) -> Ai=-0.378814 Bi=-0.198290 Ai'=0.314584 Bi'=-0.675611
Airy(-2) -> Ai=0.227407 Bi=-0.412303 Ai'=0.618259 Bi'=0.278795
Airy(-1) -> Ai=0.535561 Bi=0.103997 Ai'=-0.010161 Bi'=0.592376
Airy(0) -> Ai=0.355028 Bi=0.614927 Ai'=-0.258819 Bi'=0.448288
Airy(1) -> Ai=0.135292 Bi=1.207424 Ai'=-0.159147 Bi'=0.932436

Parameters

input argument

reference variable, into which the solution for Ai(x) is placed.

Aip

reference variable, into which the solution for Ai'(x) is placed.

reference variable, into which the solution for Bi(x) is placed.

Bip

reference variable, into which the solution for Bi'(x) is placed.

Returns

0 if successful.

Interactive Calculator

Aip

Bip

Result –