1
$\begingroup$

I read that $\int\limits_0^1 xJ_n(j_{na}x) J_n(j_{nb}x) dx={1\over 2}\delta_{ab}[J_n'(j_{na})]^2$, where $j_{na},j_{nb}$ are zeros of $J_n$, the Bessel function of the $n$th degree. Is there a simple proof for this? Thanks.

  • 0
    Where did you read this ?2011-11-04
  • 0
    @Sasha: On quite a few websites talking about Bessel functions. This demonstrates the orthogonality and helps us normalize... But I have't found a website that gives a proof of this. They tend to say "it can be shown that..." I have tried proving it myself, starting with the Bessel equations. I have managed to show that if $a\neq b$ tgen tge integral vanishes, but I can't show the $a=b$ case. Thought I might let $a=b+\epsilon$ Then take limit as epsilon tends to 0...2011-11-04
  • 0
    I'm tempted to think this can be proven as a special case of the orthogonality result in Sturm-Liouville theory: [url]http://en.wikipedia.org/wiki/Sturm–Liouville_theory[\url]. I'll leave the details to someone else though.2011-11-04
  • 0
    @Ragib I suspect that the reason why the URL you posted isn't working is that there's an en-dash in the title. I also suspect that a Wikipedia page with the same title but a hyphen rather than an en-dash is a redirect page that directs to the article whose title has an en-dash. Let's try it: [url]en.wikipedia.org/wiki/Sturm-Liouville_theory[/url]2011-11-04
  • 0
    Typo. I'll try again: [url]en.wikipedia.org/wiki/Sturm-Liouville_theory[\url]2011-11-04
  • 0
    @Michael: `[Sturm-Liouville](http://en.wikipedia.org/wiki/Sturm-Liouville_theory)` produces [Sturm-Liouville](http://en.wikipedia.org/wiki/Sturm-Liouville_theory). They use Markdown here, not BBCode. :)2011-11-04
  • 0
    Thanks guys, I need to start remembering this!2011-11-04

1 Answers 1