= Cauchy kernel fundamental solution
{c}
{title2=$\partial_{\bar z}(1/(\pi z))=\delta_0$}
The <locally integrable function> $E(z)=1/(\pi z)$ on the plane is a <fundamental solution> for the <Wirtinger derivative> $\partial_{\bar z}=(\partial_x+i\partial_y)/2$. Let $L=(4\pi)^{-1}\log(x^2+y^2)$ be the <fundamental solution of the Laplace equation>. Its first <distributional derivatives> agree with its ordinary ones: deleting a radius-$\varepsilon$ disk in <integration by parts> gives an error $O(\varepsilon|\log\varepsilon|)$, tending to zero. Hence $E=4\partial_zL$ and
$$
\partial_{\bar z}E=4\partial_{\bar z}\partial_zL=\Delta L=\delta_0.
$$
The analogous <fundamental solution> for $\partial_z$ is $1/(\pi\bar z)$. The <singular support> of each kernel is exactly $\{0\}$: they are smooth elsewhere, and smoothness near zero would make their <Wirtinger derivatives> smooth there, contradicting the <Dirac delta distribution> identity.
``
Target: Distributional Cauchy-Riemann lemma. Parent: Cauchy-Riemann equations. Intended file: analysis.bigb.
``
Back to article page
