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Dirac Delta Function

The Dirac delta function, $ \delta(t-t')$ , has the property

$\displaystyle \delta(t-t') =0$   $\displaystyle \mbox{\hspace{0.5cm}for\hspace{0.5cm}$t\neq t'$}$$\displaystyle .$ (17)

In addition, however, the function is singular at $ t=t'$ in such a manner that

$\displaystyle \int_{-\infty}^\infty \delta (t-t')\,dt' = 1.$ (18)

It follows that

$\displaystyle \int_{-\infty}^\infty f(t)\,\delta(t-t')\,dt' = f(t),$ (19)

where $ f(t)$ is an arbitrary function that is well behaved at $ t=t'$ . It is also easy to see that

$\displaystyle \delta(t'-t) = \delta(t-t').$ (20)

Richard Fitzpatrick 2014-06-27