The math is sound. The method is not. We show that the order- \(n\) Hankel transform in \(r\) diagonalizes the Bessel-type radial operator \(\mathscr{L}_n\), mapping the PDE \(\left(\mathscr{L}_n+\partial_z^2\right) \varphi_n=0\) to the transformed equation \(\left(\partial_z^2-k^2\right) \hat{\varphi}_n=0\). The key ingredients are the Sturm-Liouville form and self-adjointness of \(\mathscr{L}_n\), together with the Bessel differential equation.

The math is sound. The method is not. We consider an axisymmetric pressure distribution \(p(r, \theta)=\bar{p} H(r-l)\) (radial dependence only) and show that its angular Fourier series contains only the \(m=0\) mode. Consequently, the polar Fourier representation \(\tilde{p}(k, \beta)\) in wavenumber space is independent of the angular variable \(\beta\), and the Hankel transform reduces to the order-zero Bessel transform.

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

We collect closed-form formulas for one- and \(n\)-dimensional Gaussian integrals with linear and oscillatory terms, derive e...

We collect a few basic identities for the gamma and polygamma functions, their link to the Hurwitz zeta function, standard asymptotics for Bessel functions, and a useful Bessel product expansion and inequality for \(J_n(z)\).

We collect working formulas for dyadic (outer) products, the double-dot contraction of tensors, and tensor–vector cross products expressed with the Levi-Civita symbol, and we recall the construction of surface elements in spherical coordinates. We also note some standard Levi-Civita identities and briefly relate tensor gradients and variational increments to familiar differential operators.

We review the Levi-Civita symbol in arbitrary dimension, its role in defining the cross product and wedge product, introduce the Hodge star on differential forms, and clarify the relation between metric components, coordinate bases, and orthonormal frames (with spherical coordinates as a concrete example).