A.1 Distributions

A delta function is not a function in the normal sense. Infinity is not a proper number.

However, delta functions have a property that can be used to define them. That property is called the “filtering property.” If you multiply a delta function by a smooth function $\phi(x)$ and integrate over all $x$, you get the value of the function at the location of the delta function:

$$\int_{x=-\infty}^\infty \phi(x) \delta(x-\xi){ \rm d}x = \phi(\xi)$$

The reason is that the delta function is everywhere zero except at the single point $x=\xi$. So you can replace $\phi(x)$ by $\phi(\xi)$ without changing anything. And $\phi(\xi)$ is a constant that can be taken out of the integral.

You can reverse that statement and define the delta function as the “distribution” that produces the result above for any smooth function $\phi$. (The functions $\phi$ are normally further constrained by a requirement that they must become zero at their ends.)

In a similar way you can also define the derivative of the delta function, the dipole $\delta'$. It is the distribution for which, for any smooth $\phi$,

$$\int_{x=-\infty}^\infty \phi(x) \delta'(x-\xi){ \rm d}x = -\phi'(\xi)$$

To see why you want to define it this way, perform a formal integration by parts.