The Dirac Delta Function

The Dirac Delta function is a normalisation tool, often used so that an integral equates to some desired value. It is defined as
\[\delta (x)=lim_{\epsilon \rightarrow 0} \left\{ \begin{array}{c} \frac{1}{2 \epsilon} \; - \epsilon \lt x \lt \epsilon \\ 0 \; otherwise \end{array} \right. \]

and has the property
\[\int^{\infty}_{- \infty} \delta (x)dx=1\]
  (1)
The Dirac Delta function can be used to return a specific value for a function that appears in an integral. For example
\[f( \alpha )= \int^{\infty}_{- \infty} f(x) \delta (x- \alpha )dx\]
.

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