Spherical Bessel Function Zeros
Find the positive zeros of the spherical Bessel functions jₙ(x) or yₙ(x) in ascending order. Choose the kind, order, and count to list Newton-refined roots in a table and chart.
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Input
Choose the kind of spherical Bessel function, the order n, and how many zeros you want to list the positive zeros in ascending order.
Function kind
First kind jₙ(x)
Integer from 0 to 50
Integer from 1 to 30
Result
First zero of First kind jₙ(x) with order 0
3.1415926536
Function and zero positions
| Index | Zero x |
|---|---|
| 1 | 3.1415926536 |
| 2 | 6.2831853072 |
| 3 | 9.4247779608 |
| 4 | 12.5663706144 |
| 5 | 15.7079632679 |
How it works
- The spherical Bessel functions are evaluated with closed-form recurrences. The first kind starts from j0(x)=sin(x)/x and j1(x)=sin(x)/x^2 - cos(x)/x; the second kind from y0(x)=-cos(x)/x and y1(x)=-cos(x)/x^2 - sin(x)/x, raising the order with f(k+1)=((2k+1)/x)f(k) - f(k-1).
- Zeros are located by scanning for sign changes on a fine grid, bracketing each root, and refining it with bisection and Newton iteration. For x smaller than the order, j_n is evaluated with a power series to avoid numerical instability.
- The order n is an integer from 0 to 50 and the count is from 1 to 30. Results are accurate to roughly ten decimal places.
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