#delta function

[Click here for a PDF version of this post] Motivation I’ve been assembling a table of all the Green’s functions that can be used in electrodynamics. There’s one set of those Green’s functions left to fill in, the Green’s functions for the spacetime gradient: \begin{equation}\label{eqn:spacetimeGradientGreens:20} \lr{\spacegrad + \inv{c}\PD{t}{}} G(\Bx, \Bx’, t, t’) = \delta(\Bx – \Bx’)\delta(t –…

[Click here for a PDF version of this and related posts] We’ve now evaluated the 1D Green’s function and 2D Green’s function for the wave equation. For the sake of completeness, now let’s evaluate the Green’s function for the 3D wave equation operator. Again with \( \Br = \Bx – \Bx’, \tau = t – t’ \) we want the \( \epsilon \rightarrow 0 \) limit…

[Click here for a PDF version of this post] The Green’s function(s) \( G(\Br, \tau) \) for the 3D wave equation \begin{equation}\label{eqn:waveEquationGreens:40} \lr{ \spacegrad^2 – \inv{c^2}\frac{\partial^2}{\partial t^2} } G(\Br, \tau) = \delta(\Br) \delta(\tau), \end{equation} where \begin{equation}\label{eqn:waveEquationGreens:20} \begin{aligned} \Br &= \Bx – \Bx’ \\ r &= \Abs{\Br} \\ \tau &=…

[Click here for a PDF version of this post, and others in this series] The following recent posts explored 1D Green’s functions for the Helmholtz and Laplacian operators.  There was a sign error (wrong residue sign for a negatively oriented contour) that I made near the beginning that caused a lot of trouble.  Having found the error, I’ve now reworked all that exploratory content into a more…

[Click here for a PDF version of this post, and the previous posts in this series.] We used a complicated limiting argument to show that the \( \mathrm{sgn}(x – x’) \) factor in the contour integral derivation of the Helmholtz operator Green’s function was wrong. Having discovered, even if slightly by accident, what the correct form of that Green’s function is, we can check it more directly. This…

[Click here for a PDF version of this post, and the others in this series.] Having blundered our way to what appears to be the correct Green’s function for the 1D Helmholtz operator, let’s further validate that by deriving the Green’s function for the 1D Laplacian. We should also be able to verify that it has the correct delta function semantics. The 1D Laplacian Green’s function. Expanding the…

[Click here for a PDF version of this, and previous, posts in this series]. The discontinuity in the derived 1D Helmholtz Green’s function is somewhat surprising. Let’s try to verify that this works or find what does. The first think to check is that \begin{equation}\label{eqn:helmholtzGreens:1220} \lr{ \spacegrad^2 + k^2} G(x,x’) = 0, \end{equation} at locations where \( x \ne x’ \). Since we…

Screenshot [Click here for a PDF version of this post] Once again, I was reading my Jackson [1], which characteristically had the statement “the […] integral can easily be shown to have the value \( 4 \pi \)”, in a discussion of electrostatic energy and self energy. The integral is \begin{equation}\label{eqn:selfEnergyIntegral:20} I = \int \frac{\Brho}{\rho^3} \cdot \frac{\Brho + \Bn}{\Norm{\Brho…