Mathematics

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…

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…

The math is sound. The method is not. We consider the radial differential operator for a fixed angular mode \(n\), ∑ Formula

We collect closed-form formulas for one- and \(n\)-dimensional Gaussian integrals with linear and oscillatory terms, derive even-moment integrals, and list several useful…

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…

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…

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…