Laplacian (Example 4.22)
Let $y \in \mathbb{R}^n$ and denote
\[
G_y(x)=G_y^n(x):= \begin{cases}-\frac{1}{(n-2) \omega_{n-1}}|x-y|^{2-n} & \text { if } n \in \mathbb{N} \backslash\{2\} \\ \frac{1}{\omega_1} \log |x-y| & \text { if } n=2,\end{cases}
\]
$\Delta G_y=\delta_y \quad$ in $\mathscr{D}^{\prime}\left(\mathbb{R}^n\right)$
Cauchy-Riemann differential operators (Example 4.23)
\[
\frac{\partial}{\partial z}:=\frac{1}{2}\left(\frac{\partial}{\partial x}-\mathrm{i} \frac{\partial}{\partial y}\right), \quad \frac{\partial}{\partial \bar{z}}:=\frac{1}{2}\left(\frac{\partial}{\partial x}+\mathrm{i} \frac{\partial}{\partial y}\right)
\]
\begin{aligned}
\frac{\partial}{\partial \bar{z}}\left(\frac{1}{\pi z}\right) & =\delta_0 . \\
\frac{\partial}{\partial z}\left(\frac{1}{\pi \bar{z}}\right) & =\delta_0 .
\end{aligned}