Module 1 : A Crash Course in Vectors
Lecture 6 : Laplacian
  Dirac- Delta Function :
  In electromagnetism, we often come across use of a function known as Dirac- $\delta$ function. The peculiarity of the function is that though the value of the function is zero everywhere, other than at one point, the integral of the function over any region which includes this singular point is finite. We define
 
\begin{displaymath}\delta(x-a)= 0\ \ {\rm if}\ \ x\ne a\end{displaymath}
  with
\begin{displaymath}\int f(x)\delta(x-a) dx = f(a)\end{displaymath}
  where $f(x)$ is any function that is continuous at $x=a$, provided that the range of integration includes the point $x=a$. Strictly speaking, $\delta(x-a)$ is not a function in the usual sense as Riemann integral of any function which is zero everywhere, excepting at discrete set of points should be zero. However, one can look at the $\delta$ function as a limit of a sequence of functions. For instance, if we define a function $\delta_\epsilon(x)$ such that
 
\begin{eqnarray*} \delta_\epsilon(x) &=& \frac{1}{\epsilon}\ \ {\rm for}\ \ -\fr... ...lon}{2}\\ &=& 0\ \ {\rm for}\ \ \mid x\mid > \frac{\epsilon}{2} \end{eqnarray*}
  Then $\delta(x)$ can be thought of as the limit of $\delta_\epsilon(x)$ as $\epsilon\rightarrow 0$.
   
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