Alternatives to Dirac's Delta Function

The Dirac Delta function is often represented as an infinite spike of thickness, a limit function of a rectangle of area 1..
\[\delta (x)=lim_{\epsilon \rightarrow 0} \left\{ \begin{array}{c} \frac{1}{2 \epsilon} \; - \epsilon \lt x \lt \epsilon \\ 0 \; otherwise \end{array} \right. \]

This function is not continuous or differentiable at  
. We can find smoother functions with the same desirable properties as the Dirac Delta function, of having area equal to one, and appearing in an integral to define a function at a point. Examples are
\[\delta_1=\frac{n}{\sqrt{\pi}} e^{-n^2x^2}\]

\[\delta_2=\frac{n}{\pi} \frac{1}{1+n^2x^2}\]

\[\delta_3=\frac{sin nx}{\pi x}= \frac{1}{2 \pi} \int^n_{-n} e^{ixt} dt\]

Add comment

Security code