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

\[ \delta(\vb x) = \lim_{\epsilon \to 0} h_{\epsilon}(\vb x) = \begin{cases} \infty & \qif \vb x = \vb 0 \\ 0 & \qif \vb x \ne \vb 0 \end{cases} \] with the function \[ h_{\epsilon}(\vb x) = \begin{cases} \frac{3}{4 \pi \epsilon^3} & \qif |\vb x| \le \epsilon \\ 0 & \qif |\vb x| \gt \epsilon \end{cases} \] which describes a unit mass distributed over a small sphere of radius \(\epsilon\).

\(\delta(\vb x)\) is the limit of the sequence of functions \(h_\epsilon(\vb x)\) for \(\epsilon \to 0\).

We are now able to evaluate the density of the sphere as \(\rho(\vb x) = m h_\epsilon(\vb x)\).

Further, the integral over a domain \(G\) containing the point \(\vb x\) is finite, i.e.,

\[ \int_G h_\epsilon(\vb x) \, \dd {\vb x} = 1. \]

We calculate the weak limit of the function sequence \(h_\epsilon(\vb x)\) for \(\epsilon \to 0\) and any continuous test function \(\Phi(\vb x)\)

\[ \lim_{\epsilon \to 0} \int \Phi(\vb x) h_\epsilon(\vb x) \, \dd {\vb x} = \Phi(0) \]

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

It holds \[ \int_G \delta(\vb x) \, \dd {\vb x} = 1 \qif \vb x = \vb 0 \in G. \]