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Archive for 2010年2月

## Addition formula for half-integer order Bessel function?

2010年02月24日 留下评论

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.

$\displaystyle\lim_{m\rightarrow \infty}\left[\left(\frac{1}{m}\right)^m+\left(\frac{2}{m}\right)^m+\cdots+\left(\frac{m-1}{m}\right)^m\right]=\frac{1}{e-1}$