The Dirac Delta function is denoted by $\delta(x)$.

Consider any function $\Delta(x)$ that has the following properties

• Total area of $1$
• A characteristic width given by the parameter $w$

Some examples:

• Gaussian: $\frac{1}{\sqrt{2\pi w^2}}\exp(-x^2/2w^2)$
• sinc function: $\frac1w\text{sinc}(\pi x/w)$
  • $\text{sinc}(x)\equiv\sin(x)/x$
• Rectangular pulse of width $w$ and height $h=\frac1w$
• Triangle Pulse of width $w$ and height $h=2/w$

The dirac delta is then defined by

$$ \delta(x)=\lim_{w\to 0}\Delta(x) $$

In every case in the limit we get a spike with the following properties

• zero width
• infinite height
• an area of $1$

$\delta(x)$ really only makes mathematicall (and is only useful) under integration

We can translate $\delta(x)$ to center the spike at a point $x_0$→$\delta(x-x_0)$