Bessel Function Zeros Calculator
Find the positive zeros of the Bessel functions Jn(x) and Yn(x) in ascending order. Choose the kind, order, and count to see the results in a graph and a table.
Input
Find the positive zeros of the Bessel function Jn(x) or Yn(x) in ascending order. Choose the kind, the order n, and how many to list.
Kind
Integer from 0 to 20
From 1 to 50
Result
First positive zero of J0
2.4048255577
Graph of J0 and its zeros
List of positive zeros of J0
| Index | Zero of J0 |
|---|---|
| 1 | 2.4048255577 |
| 2 | 5.5200781103 |
| 3 | 8.6537239116 |
| 4 | 11.7915335521 |
| 5 | 14.9309174308 |
How it works
- Finds the positive zeros of the Bessel function of the first kind Jn(x) or the second kind Yn(x), listed in ascending order.
- Initial guesses come from the McMahon asymptotic expansion and are then refined with Newton iteration, with bisection as a safe fallback if the step leaves the interval.
- The order n is an integer from 0 to 20 and the count ranges from 1 to 50. Higher orders are limited to preserve numerical accuracy.
- Bessel function values are evaluated with a power series for small x and with backward recurrence (Miller algorithm) or asymptotic expansions otherwise.
- Zeros of Bessel functions appear as eigenvalues in many physics problems, such as vibrations of a circular membrane, waveguide modes, and heat conduction.
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Bessel Function Zeros Calculator