Hey guys I was wondering if you could help me out with a proof of the recursion relations of Bessel functions on my homework:(adsbygoogle = window.adsbygoogle || []).push({});

Show by direct differentiation that

[tex]

J_{\nu}(x)=\sum_{s=0}^{\infty} \frac{(-1)^{s}}{s!(s + \nu)!} \left (\frac{x}{2}\right)^{\nu+2s}

[/tex]

obeys the important recursion relations

[tex]

J_{\nu-1}(x)+J_{\nu+1}(x) = \frac{2\nu}{x}J_{\nu}(x)

[/tex]

[tex]

J_{\nu-1}(x)-J_{\nu+1}(x) = 2J_{\nu}(x)

[/tex]

I've tried differentiating with respect to x but I get a factor of 2s that's no good. And I've also tried replacing nu with nu plus one and nu minus one but that ends up with a lot of s terms as well. I am pretty much lost on what to do so if you could just point me in the right direction that'd be great. Thanks a lot.

Csmines

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# Homework Help: Bessel Functions

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