**Definition 0.0.1 **  The **Green's function** denoted $G$ for a linear operator $\mathcal{L}$ is the solution to the following equation where $\delta$ is the [Dirac delta](https://kaedon.net/l/^wna1).
\[\mathcal{L}G(\vec{r},\vec{r}')=-4\pi\delta(\vec{r}-\vec{r}')\]