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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.

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An elegant result on sum of powers

2010年02月21日 留下评论

\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}

Here it is. It is highly non-trivial that the left hand side, involving sum of powers of integers, yields something like e. Elegant isn’t it.  To prove it you need Euler-Maclaurin formula which is said to be one of the indispensable tools in any mathematitian’s bag of tricks but not taught in any undergraduate course – regardless of its elementary nature, nothing more advanced than calculus. The proof can be found here

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