In a Green's function \(G(x,x')\), the primed variable \(x'\) marks the source point — the location of a unit impulse — while \(x\) is the field point where the response is measured. This can be seen as a matrix element \(\langle x | L^{-1} | x' \rangle\) of the inverse operator. For the 1D Poisson equation \(-\frac{d^2 u}{dx^2}=f\), we obtain \(G(x,x')=-\frac12|x-x'|\) using \(\frac{d^2}{dx^2}|x-x'|=2\delta(x-x')\). Additional examples illustrate the same concept.

浓雾氤氲的森林，分不清是清晨还是黄昏，天边应该是有丝丝破碎的晨曦亦或是夕阳，还未走进你眼眸，便融化进了昏霭，染成一片模糊的光晕。你的思绪也仿若虚无之地，白茫茫一片，犹豫，抬起脚，犹豫，犹豫，犹豫，向前？向后？是追忆还是邂逅迷茫？

We show how the spectral relation \(\partial_z \widehat{\varphi}(k, 0)=k \tanh (k h) \widehat{\varphi}(k, 0)\) implies the physical-space operator identity \(\varphi_z(\cdot, \cdot, 0)=(|D| \tanh (h|D|)) \varphi(\cdot, \cdot, 0)\), by inverse Fourier transform and the commutation of \(\partial_z\) with the horizontal Fourier transform.

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 factor the quadratic denominator \(a+b \cos \gamma+c \cos ^2 \gamma\) and reduce the computation of its Fourier coefficients to those of \(1 /(\cos \gamma-\lambda)\). Solving the associated second-order recurrence for the Fourier coefficients yields a closed-form expression in terms of \(\lambda_{ \pm}\)and \(q_{ \pm}=\lambda_{ \pm}+i \sqrt{1-\lambda_{ \pm}^2}\), leading to the final explicit formula for \(c_m(k)\) in the steady ship-wave setting.