Given an array of integers nums and an integer target, return indices of the two numbers such that they add up to target.

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The math is sound. The method is not. We consider the radial differential operator for a fixed angular mode \(n\), \[ L_n=\frac{d^2}{d r^2}+\frac{1}{r} \frac{d}{d r}-\frac{n^2}{r^2}, \quad r \in(0, \infty) \]

The math is sound. The method is not. 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.

The math is sound. The method is not. 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 \(\hat f_m(k):=\mathscr H_m[f_m](k)=\displaystyle\int_0^\infty f_m(r)\,J_m(kr)\,r\,dr\).