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I am trying to verify the following formula involving Bessel functions of the first kind and am having no luck. The formula is

$$ \int{\omega} J_n(\rho \omega)\mathrm d\omega = \frac {1} {\rho} \left\{ -\omega J_{n-1} (\rho \omega) + n \int{J_{n-1}(\rho \omega)\mathrm d\omega } \right\} $$

I apologize if this is painfully obvious with integration by parts but I couldn't see it. Moreover, I get the impression from this other post about a nearly identical integral that the above may not be right.

Any help is greatly appreciated. Also, if there is a simpler way to express/solve this integral, I would also be very grateful for that.

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    You can set $\rho=1$ without loss of generality. More importantly: did you try to prove all these relations at once, relying on the generating functions of the Bessel functions of the first kind?2011-05-06
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    I'm sorry, but I don't follow. I am not aware of any generating function for J_n.2011-05-06
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    See the Laurent series here: http://en.wikipedia.org/wiki/Bessel_function#Properties2011-05-06
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    Again I am sorry, perhaps I am missing something obvious, but that is not a generating function for J_n. Rather, it's a generating function for $e^{(x/2)(t-1/t)}$ that uses J_n(x) as it's coefficients.2011-05-07
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    The idea was to sum over $n$ the relations you tried to prove multiplied by $t^n$ and to see what could be said about the resulting *generating* function. However, see below.2011-05-07
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    You can make the trivial substitution $u=\rho\omega$ and find that your integral now has a factor of $\frac1{\rho}$. Then my answer in that other question applies.2011-05-07

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One can set $\rho=1$ without loss of generality. According to this page (see the paragraph $p+1$ dependency), $$ \omega J_{n}(\omega)=(n-1)J_{n-1}(\omega)-\omega (J_{n-1})'(\omega)=-(\omega J_{n-1}(\omega))'+nJ_{n-1}(\omega). $$ Hence a primitive of $\omega J_{n}(\omega)$ is $-\omega J_{n-1}(\omega)$ plus $n$ times a primitive of $J_{n-1}(\omega)$. This is your formula.

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    @anon: the formula Didier quotes is also the second formula in [formula 10.6.2 in the DLMF](http://dlmf.nist.gov/10.6.E2); replace $\nu$ with $n-1$ and $\mathscr{C}$ with $J$.2011-05-07
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    Thank you both. You have to please excuse my daftness. It has been quite a while since I took calculus and I have never really had to deal with the special functions before. Thanks again.2011-05-07