Airy Function Zeros Calculator
Find the negative real zeros aₙ and bₙ of the Airy functions Ai(x) and Bi(x). Computed via asymptotic estimates refined with Newton's method, shown in a table.
Input
Pick the Airy function and how many zeros to find, and the negative real zeros are listed for you.
Function
First kind Ai(x)
From 1 up to 12
Result
Zero number 1: aₙ
-2.3381074105
aₙ and the computed zeros
| Index n | Zero aₙ | Derivative at zero |
|---|---|---|
| 1 | -2.3381074105 | 7.012e-1 |
| 2 | -4.0879494441 | -8.031e-1 |
| 3 | -5.5205598281 | 8.652e-1 |
| 4 | -6.7867080901 | -9.109e-1 |
| 5 | -7.9441335871 | 9.473e-1 |
| 6 | -9.022650853 | -9.779e-1 |
| 7 | -10.0401743442 | 1.004e+0 |
| 8 | -11.0085242273 | -1.028e+0 |
How it works
- The Airy functions Ai(x) and Bi(x) both solve the differential equation y''=xy, and all of their real zeros lie on the negative real axis.
- The n-th zero uses the asymptotic estimates aₙ≈-T(3π(4n-1)/8) and bₙ≈-T(3π(4n-3)/8) as starting guesses.
- Each guess is refined with Newton's method, evaluating Ai(x), Bi(x) and their derivatives from Maclaurin series.
- The table lists each zero x together with the derivative value at that point (Ai'(aₙ) or Bi'(bₙ)).
- The series loses precision for large magnitudes, so the number of zeros supported has an upper limit.
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Airy Function Zeros Calculator