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Here is a summary of Green’s functions for a number of gradient related differential operators (many of which are of interest for electrodynamics, and most of them have been derived recently in blog posts.) These Green’s functions all satisfy
\begin{equation}\label{eqn:deltaFunctions:120}
\delta(\Bx – \Bx’) = L G(\Bx, \Bx’).
\end{equation}

Let \( \Br = \Bx – \Bx’ \), \( r = \Norm{\Br} \), \( \mathbf{\hat{r}} = \Br/r \), and \( \tau = t – t’ \), then

1. Gradient operator, \( L = \spacegrad \), in 1D, 2D and 3D respectively
   \begin{equation}\label{eqn:deltaFunctions:25}
   \begin{aligned}
   G\lr{ \Bx, \Bx’ } &= \frac{\mathbf{\hat{r}}}{2} \\
   G\lr{ \Bx, \Bx’ } &= \frac{1}{2 \pi} \frac{\mathbf{\hat{r}}}{r} \\
   G\lr{ \Bx, \Bx’ } &= \inv{4 \pi} \frac{\mathbf{\hat{r}}}{r^2}.
   \end{aligned}
   \end{equation}

2. Laplacian operator, \( L = \spacegrad^2 \), in 1D, 2D and 3D respectively
   \begin{equation}\label{eqn:deltaFunctions:20}
   \begin{aligned}
   G\lr{ \Bx, \Bx’ } &= \frac{r}{2} \\
   G\lr{ \Bx, \Bx’ } &= \frac{1}{2 \pi} \ln r \\
   G\lr{ \Bx, \Bx’ } &= -\frac{1}{4 \pi r}.
   \end{aligned}
   \end{equation}

3. Second order Helmholtz operator, \( L = \spacegrad^2 + k^2 \) for 1D, 2D and 3D respectively
   \begin{equation}\label{eqn:deltaFunctions:60}
   \begin{aligned}
   G\lr{ \Bx, \Bx’ } &= \pm \frac{1}{2 j k} e^{\pm j k r} \\
   G(\Bx, \Bx’) &= \frac{1}{4 j} H_0^{(1)}(\pm k r) \\
   G\lr{ \Bx, \Bx’ } &= -\frac{1}{4 \pi} \frac{e^{\pm j k r }}{r}.
   \end{aligned}
   \end{equation}

4. First order Helmholtz operator, \( L = \spacegrad + j k \), in 1D, 2D and 3D respectively

   \begin{equation}\label{eqn:deltaFunctions:80}
   \begin{aligned}
   G\lr{ \Bx, \Bx’ } &= \frac{j}{2} \lr{ \mathbf{\hat{r}} \mp 1 } e^{\pm j k r} \\
   G\lr{ \Bx, \Bx’ } &= \frac{k}{4} \lr{ \pm j \mathbf{\hat{r}} H_1^{(1)}(\pm k r) – H_0^{(1)}(\pm k r) } \\
   G\lr{ \Bx, \Bx’ } &= \frac{e^{\pm j k r}}{4 \pi r} \lr{ jk \lr{ 1 \mp \mathbf{\hat{r}} } + \frac{\mathbf{\hat{r}}}{r} }.
   \end{aligned}
   \end{equation}

   This is also the Green’s function for a left acting operator \( G(\Bx, \Bx’) \lr{ – \lspacegrad + j k } = \delta(\Bx – \Bx’) \).

5. Wave equation, \( \spacegrad^2 – (1/c^2) \partial_{tt} \), in 1D, 2D and 3D respectively
   \begin{equation}\label{eqn:deltaFunctions:140}
   \begin{aligned}
   G(\Br, \tau) &= -\frac{c}{2} \Theta( \pm \tau – r/c ) \\
   G(\Br, \tau) &= -\inv{2 \pi \sqrt{ \tau^2 – r^2/c^2 } } \Theta( \pm \tau – r/c ) \\
   G(\Br, \tau) &= -\inv{4 \pi r} \delta( \pm \tau – r/c ),
   \end{aligned}
   \end{equation}
   The positive sign is for the retarded solution, and the negative for advancing.

6. Spacetime gradient \( L = \spacegrad + (1/c) \partial_t \), satisfying \( L G(\Bx – \Bx’, t – t’) = \delta(\Bx – \Bx’) \delta(t – t’) \), in 1D, 2D, and 3D respectively
   \begin{equation}\label{eqn:deltaFunctions:100}
   \begin{aligned}
   G(\Br, \tau)
   &= \inv{2} \lr{ \mathbf{\hat{r}} \pm 1 } \delta(\pm \tau – r/c) \\
   G(\Br, \tau)
   &=
   \frac{
   \lr{\tau^2 – r^2/c^2}^{-3/2}
   }{2 \pi c^2}
   \lr{
   c \lr{ \mathbf{\hat{r}} \pm 1 }
   \lr{\tau^2 – r^2/c^2}
   \delta(\pm \tau – r/c)
   -\lr{ \Br + c \tau }
   \Theta(\pm \tau – r/c)
   }
   \\
   G(\Br, \tau)
   &= \inv{4 \pi r} \delta(\pm \tau – r/c)
   \lr{
   \frac{\mathbf{\hat{r}}}{r}
   +
   \lr{ \mathbf{\hat{r}} \pm 1} \inv{c} \PD{t’}{}
   }
   \end{aligned}
   \end{equation}
   The plus sign is for the retarded solution, and negative for advanced.