$$\int z^{l+1}J_m(z)dz=z^{l+1}J_{m+1}(z)+(l-m)z^lJ_m(z)-(l^2-m^2)\int z^{l-1}J_m(z)dz$$ Where $J_m$ is a Bessel function of the $mth$ kind. I know it is integration by parts but I am not sure exactly what they are setting there $u$s and $v$s
2026-03-29 15:59:59.1774799999
How can I prove this identity for the bessel function?
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This is an application of the formula DLMF 10.6.6 for the derivative of ordinary Bessel functions. In your special case it states: $$\frac{d}{dz}\left(z^\nu J_\nu(z)\right) = z^\nu J_{\nu -1}(z).$$
I would also point out that $J_m(z)$ is not a Bessel function of the $m$-th kind, but a Bessel function of the first kind of order $m$. The Digital Library of Mathematical Functions offers more information about them.