Editing Dirac delta function (section)

=====The Poisson kernel=====
The [[Poisson kernel]]
<math display="block">\eta_\varepsilon(x) = \frac{1}{\pi}\mathrm{Im}\left\{\frac{1}{x-\mathrm{i}\varepsilon}\right\}=\frac{1}{\pi} \frac{\varepsilon}{\varepsilon^2 + x^2}=\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i} \xi x-|\varepsilon \xi|}\,d\xi</math>

is the fundamental solution of the [[Laplace equation]] in the upper half-plane.{{sfn|Stein|Weiss|1971|loc=§I.1}}  It represents the [[electrostatic potential]] in a semi-infinite plate whose potential along the edge is held at fixed at the delta function. The Poisson kernel is also closely related to the [[Cauchy distribution]] and [[Kernel (statistics)#Kernel functions in common use|Epanechnikov and Gaussian kernel]] functions.<ref>{{Cite book|last=Mader|first=Heidy M.|url={{google books |plainurl=y |id=e5Y_RRPxdyYC}}|title=Statistics in Volcanology|date=2006|publisher=Geological Society of London|isbn=978-1-86239-208-3|language=en|editor-link=Heidy Mader|page=[{{google books |plainurl=y |id=e5Y_RRPxdyYC|page=81}} 81]}}</ref> This semigroup evolves according to the equation
<math display="block">\frac{\partial u}{\partial t} = -\left (-\frac{\partial^2}{\partial x^2} \right)^{\frac{1}{2}}u(t,x)</math>

where the operator is rigorously defined as the [[Fourier multiplier]]
<math display="block">\mathcal{F}\left[\left(-\frac{\partial^2}{\partial x^2} \right)^{\frac{1}{2}}f\right](\xi) = |2\pi\xi|\mathcal{F}f(\xi).</math>