Spherical Bessel Function Calculator

Enter an order n and x to evaluate the spherical Bessel functions of the first kind jₙ(x) and second kind yₙ(x), with neighboring orders shown in a table.

Input

Enter the order n (a non-negative integer) and the argument x to evaluate the spherical Bessel functions jₙ(x) and yₙ(x).

Order n

Non-negative integer up to 100

Argument x

yₙ(x) is defined only when x is greater than zero.

Result

First kind j2(5)

0.1347312101

Second kind y2(5)

0.1649954576

First kind j3(x)

0.2298206182

Neighboring orders

Function	Value
j1(x)	-0.0950894081
j2(x)	0.1347312101
j3(x)	0.2298206182
y2(x)	0.1649954576
y3(x)	-0.0154429099

Curve of j2(x)

How it works

The first-kind series starts from j0(x)=sin x / x and j1(x)=sin x / x^2 − cos x / x.
The second-kind series starts from y0(x)=−cos x / x and y1(x)=−cos x / x^2 − sin x / x.
The three-term recurrence f(n+1) = ((2n+1)/x)·f(n) − f(n−1) steps the order up.
The second kind yn is computed by upward recurrence, which is stable for x greater than zero.
The first kind jn uses upward recurrence when the order n is below x and backward (Miller) recurrence when n is at least x, normalizing with j0 to keep accuracy.
yn(x) diverges and is undefined for x at or below zero, so it cannot be evaluated there.

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