I am attempting to integrate a Bessel function of the first kind multiplied by a linear term: $\int xJ_n(x)\mathrm dx$ The textbooks I have open in front of me are not useful (Boas, Arfken, various Schaum's) for this problem. I would like to do this by hand. Is it possible? I have had no luck with expanding out $J_n(x)$ and integrating term by term, as I cannot collect them into something nice at the end.
If possible and I just need to try harder (i.e. other methods or leaving it alone for a few days and coming back to it) that is useful information.
Thanks to anyone with a clue.
Best Answer
At the very least, $\int u J_{2n}(u)\mathrm du$ for integer $n$ is expressible in terms of Bessel functions with some rational function factors.
To integrate $u J_0(u)$ for instance, start with the Maclaurin series:
$$u J_0(u)=u\sum_{k=0}^\infty \frac{(-u^2/4)^k}{(k!)^2}$$
and integrate termwise
$$\int u J_0(u)\mathrm du=\sum_{k=0}^\infty \frac1{(k!)^2}\int u(-u^2/4)^k\mathrm du$$
to get
$$\int u J_0(u)\mathrm du=\frac{u^2}{2}\sum_{k=0}^\infty \frac{(-u^2/4)^k}{k!(k+1)!}$$
thus resulting in the identity
$$\int u J_0(u)\mathrm du=u J_1(u)$$
For $\int u J_2(u)\mathrm du$, we exploit the recurrence relation
$$u J_2(u)=2 J_1(u)-u J_0(u)$$
and
$$\int J_1(u)\mathrm du=-J_0(u)$$
(which can be established through the series definition for Bessel functions) to obtain
$$\int u J_2(u)\mathrm du=-u J_1(u)-2J_0(u)$$
and in the general case of $\int u J_{2n}(u)\mathrm du$ for integer $n$, repeated use of the recursion relation
$$J_{n-1}(u)+J_{n+1}(u)=\frac{2n}{u}J_n(u)$$
as well as the additional integral identity
$$\int J_{2n+1}(u)\mathrm du=-J_0(u)-2\sum_{k=1}^n J_{2k}(u)$$
should give you expressions involving only Bessel functions.
On the other hand, $\int u J_{\nu}(u)\mathrm du$ for $\nu$ not an even integer cannot be entirely expressed in terms of Bessel functions; if $\nu$ is an odd integer, Struve functions are needed ($\int J_0(u)\mathrm du$ cannot be expressed solely in terms of Bessel functions, and this is where the Struve functions come in); for $\nu$ half an odd integer, Fresnel integrals are needed, and for general $\nu$, the hypergeometric function ${}_1 F_2\left({{}\atop b}{a \atop{}}{{}\atop c}\mid u\right)$ is required.