Modified Spherical Bessel Function iₙ(x), kₙ(x) Calculator
Enter an order n and x to compute the modified spherical Bessel functions of the first kind iₙ(x) and second kind kₙ(x), with a table of neighboring orders and a graph.
Input
Enter an order n and an argument x (positive real) to compute the modified spherical Bessel functions iₙ(x) and kₙ(x).
Integer between 0 and 50
Positive real number (x ＞ 0)
Result
First kind iₙ(x) (n = 2, x = 2)
0.3518560886
Second kind kₙ(x) (n = 2, x = 2)
0.3454492694
Order n
2
Argument x
2
Graph of iₙ(x) (n = 2)
Values for neighboring orders
| Order n | iₙ(x) | kₙ(x) |
|---|---|---|
| 0 | 1.8134302039 | 0.1062920829 |
| 1 | 0.9743827436 | 0.1594381243 |
| 2 | 0.3518560886 | 0.3454492694 |
| 3 | 0.0947425222 | 1.0230612979 |
| 4 | 0.0202572609 | 3.926163812 |
How it works
- The first kind starts from the closed forms i0(x)=sinh(x)/x and i1(x)=(x cosh(x)−sinh(x))/x^2.
- The second kind starts from k0(x)=(π/2)e^(−x)/x and k1(x)=(π/2)e^(−x)(1+x)/x^2.
- Both use the three-term recurrence f(n−1)−f(n+1)=(2n+1)/x times f(n).
- kₙ is computed by upward recurrence (increasing order), which is numerically stable.
- iₙ is computed by backward recurrence (Miller method) and normalized with i0(x), since upward recurrence is unstable.
- Enter a positive real x and an integer order n between 0 and 50.
Reviews
Tell us what you think of this calculator.
Write a review
- Home
Modified Spherical Bessel Function iₙ(x), kₙ(x) Calculator