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Addition formula for half-integer order Bessel function?

2010年02月24日 留下评论 Go to comments

I need to know the following sum

\displaystyle\sum_{m=0}^{\infty}J_{m+1/2}(kr)J_{m+1/2}(kr')\cos(m+1/2)\theta

Well, there is indeed an addition formula of this sort for spherical, or half-integer Bessel functions in textbooks. But since spherical Bessel functions naturally emerge in three dimensions, the angle part there is a Legendre polynomial:

\displaystyle \frac{\sin kR}{kR}=\sum_{n=0}^{\infty}(2n+1)\frac{J_{n+1/2}(kr)}{\sqrt{kr}}\frac{J_{n+1/2}(kr')}{\sqrt{kr'}}P_n(\cos\theta)

where R=\sqrt{r^2+{r'}^2-2rr'\cos\theta}.

Actually the sum can be done and result is a nontrivial integral. I guess there’s no closed form for that.

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