Airy Derivative Zeros Calculator
Find the negative real zeros of the Airy derivative Ai'(x) or Bi'(x). Enter how many you need and get them in ascending order, computed with asymptotic estimates refined by Newton's method, shown in a table and graph.
Input
Choose the type and how many you need to compute the negative real zeros of the Airy derivative Ai''(x) or Bi''(x) in ascending order.
Type
Zeros of Ai''(x)
Enter an integer from 1 to 30.
Result
First zero of Zeros of Ai''(x)
-1.0187929716
Derivative and zero locations
| Index n | Zero value |
|---|---|
| 1 | -1.0187929716 |
| 2 | -3.2481975822 |
| 3 | -4.8200992112 |
| 4 | -6.1633073556 |
| 5 | -7.3721772551 |
How it works
- The Airy functions Ai(x), Bi(x) and their derivatives Ai'(x), Bi'(x) are evaluated with power series that converge around the origin. Two basic series f(x) and g(x) are summed by updating each term recursively, then combined with the constants Ai(0) and Ai'(0) to build Ai' and Bi'.
- Initial estimates of the zeros come from an asymptotic approximation. For the n-th negative zero of Ai'(x) it uses z=3*pi*(4n-3)/8, and for Bi'(x) it uses z=3*pi*(4n-1)/8, taking z^(2/3) with correction terms and flipping the sign.
- Each estimate is refined with Newton's method. The slope at a zero of the derivative uses Ai''(x)=x*Ai(x), which follows from the Airy differential equation (and likewise for Bi).
- All real solutions of Ai'(x)=0 and Bi'(x)=0 lie in the negative region. Results are listed by increasing absolute value and shown in a table and graph.
- You can request from 1 to 30 zeros. The power series converges everywhere, but rounding grows for large magnitudes, so the design keeps accuracy adequate within the displayed digits.
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Airy Derivative Zeros Calculator