Calculator
Example Data Table
| x | Ai(x) | Bi(x) | Ai'(x) | Bi'(x) |
|---|---|---|---|---|
| -5.00 | 3.50761009e-1 | 1.10265873e-1 | 3.27192818e-1 | 7.07846044e-1 |
| -1.00 | 5.35560883e-1 | 3.78537813e-1 | -1.01605671e-2 | 3.89050878e-1 |
| 0.00 | 3.55028054e-1 | 6.14926627e-1 | -2.58819404e-1 | 1.49429452e-1 |
| 1.00 | 1.35292416e-1 | 8.83060176e-1 | -1.59147441e-1 | 5.29740138e-1 |
| 5.00 | 1.08344361e-4 | 4.38528092e+2 | -2.47414037e-4 | 9.57212577e+2 |
Values are generated by the same numerical method used by the calculator.
Formula Used
The Airy functions are two independent solutions of the Airy equation: y''(z) - z y(z) = 0, or equivalently y''(z) = z y(z).
This calculator solves the first-order system: y1' = y2 and y2' = z y1, using a fourth-order Runge-Kutta integrator along a straight path from 0 to the target z.
Initial conditions at z = 0 use known constants involving the Gamma function: Ai(0), Ai'(0), Bi(0), Bi'(0). A Wronskian check compares Ai*Bi' - Ai'*Bi against 1/pi.
How to Use
- Enter x and optional imaginary part y.
- Choose a step count; try 2000 as a baseline.
- Enable adaptive refinement for tighter error control.
- Submit to compute Ai, Bi, and derivatives.
- Use the export buttons to download your results.
For large magnitudes, increase steps or enable refinement.