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I was wondering if there is a known closed form solution for the zeros of the Spherical Bessel Functions. While doing a Quantum assignment I came across them as a solution for the spherical infinite potential well. However, I only read about them as just a sequence of numbers, but no generating function or closed form expression. Any suggestions would be great.


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Have you already had a look at Watson's Treatise (cfr.…)? – Pacciu Feb 3 '12 at 1:01
@Pacciu Thanks, I just picked up a copy, and made a list of some books from that thread ;) – kηives Feb 3 '12 at 14:07

1 Answer 1

up vote 3 down vote accepted

For $n=-1,0$, finding the roots of the spherical Bessel functions $j_n(x)$ and $y_n(x)$ is somewhat easy, since:

$$\begin{array}{ll} j_{-1}(x)&=&\frac{\cos\,x}{x}&\quad&y_{-1}(x)&=&\mathrm{sinc}(x)\\ j_0(x)&=&\mathrm{sinc}(x)&\quad&y_0(x)&=&-\frac{\cos\,x}{x}\\ \end{array}$$

where $\mathrm{sinc}(x)=\dfrac{\sin\,x}{x}$ is the sine cardinal. Solving for zeros of other orders results in rather complicated transcendental equations, which I doubt have closed-form solutions. However, you will want to see these DLMF entries for some more information that can help you in numerically determining the zeros (e.g., asymptotic expansions); approximations derived from formulae there can then be subsequently polished with Newton-Raphson or some other iterative method of choice.

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