We consider the radial differential operator for a fixed angular mode \(n\), \[ L_n=\frac{d^2}{d r^2}+\frac{1}{r} \frac{d}{d r}-\frac{n^2}{r^2}, \quad r \in(0, \infty) \]

We introduce azimuthal Fourier expansion for functions with circular symmetry: we write \(f(r, \phi)\) as a sum of angular modes \(\hat{f}_n(r) e^{i n \phi}\), where \(\hat{f}_n(r)\) is obtained by integrating over \(\phi\) and measures the strength of mode \(n\) at radius \(r\). We then contrast this discrete angular Fourier series on \([0,2 \pi]\) with the usual continuous Fourier transform on \((-\infty, \infty)\), and we briefly note that this separation of radial and angular dependence is especially useful in optics, quantum mechanics, fluid dynamics, and image processing.

Hankel–Bessel Ladder Identities Claim. For suitable \(f_m(r)\) and \(k>0\), \[ \mathscr{H}_{m+1}\!\left[\left(\partial_r-\frac{m}{r}\right) f_m\right](k) =-\,k\,\hat{f}_m(k). \tag{1} \] Here \(\hat f_m(k):=\mathscr H_m[f_m](k)=\displaystyle\int_0^\infty f_m(r)\,J_m(kr)\,r\,dr\).

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basic concept ninja ninja is a build system—a tool that actually runs the compile and link commands for a software project. T...