3  Point mass

We introduce point sources.

However, the difficulty is that, e.g., for a point mass there is no proper definition of a density in the sense of a mass per volume.

Difficulties arise when we have to integrate the Poisson’s equation with point sources.

To this end, we introduce a generalized function with the desired property that it provides finite integral values for unbounded integrands.

We define

$$\delta (\mathbf{x})=\underset{ϵ\to 0}{lim}{h}_{ϵ}(\mathbf{x})=\{\begin{array}{ll}\mathrm{\infty }& \text{if}\mathbf{x}=\mathbf{0}\\ 0& \text{if}\mathbf{x}\ne \mathbf{0}\end{array}$$ with the function $$h_{ϵ}(\mathbf{x})=\{\begin{array}{ll}\frac{3}{4\pi {ϵ}^3}& \text{if}|\mathbf{x}|\le ϵ\\ 0& \text{if}|\mathbf{x}|>ϵ\end{array}$$ which describes a unit mass distributed over a small sphere of radius $ϵ$.

$\delta (\mathbf{x})$ is the limit of the sequence of functions $h_{ϵ}(\mathbf{x})$ for $ϵ\to 0$.

We are now able to evaluate the density of the sphere as $\rho (\mathbf{x})=m{h}_{ϵ}(\mathbf{x})$.

Further, the integral over a domain $G$ containing the point $\mathbf{x}$ is finite, i.e.,

$${\int }_G{h}_{ϵ}(\mathbf{x})\mathrm{d}\mathbf{x}=1.$$

We calculate the weak limit of the function sequence $h_{ϵ}(\mathbf{x})$ for $ϵ\to 0$ and any continuous test function $\mathrm{\Phi }(\mathbf{x})$

$$\underset{ϵ\to 0}{lim}\int \mathrm{\Phi }(\mathbf{x})h_{ϵ}(\mathbf{x})\mathrm{d}\mathbf{x}=\mathrm{\Phi }(0)$$

The functional which assigns to each continuous function $\mathrm{\Phi }(\mathbf{x})$ its value at the point $\mathbf{x}=0$, i.e., $\mathrm{\Phi }(0)$, is called the weak limit of the function $h_{ϵ}(\mathbf{x}),ϵ\to 0$. This functional can be used to define the density $\delta (\mathbf{x})$.

It holds $${\int }_G\delta (\mathbf{x})\mathrm{d}\mathbf{x}=1\text{if}\mathbf{x}=\mathbf{0}\in G.$$