Meaning of a Dirac function

Question

My question is about the $\delta$ function. It has following property: $$\int_{-\infty}^\infty f(x)\delta (x-t) \,\mathrm{d} x = f(t) $$ What's the meaning of the equation? Why not directly calculate $f(t)$?

Answer 1

A Dirac-delta function, $\delta(x)$, is strictly speaking not a function. It is a distribution and the equation you have above is actually a defining property of the Dirac-delta function - it only makes mathematical sense under an integral.

What the equation intuitively

$\int_{-\infty}^{\infty} f(x) \delta(x - t) dx = f(t)$

means is that $\delta(x)$ vanishes identically everywhere expect at the origin, where it is infinitely peaked.

The equation isn't used to find $f(x)$; rather, it tells you how a $\delta$-function affects $f(x)$ when integrated against it.

$\delta$-functions are extremely useful and show up everywhere in physics and mathematics, for instance, when solving certain differential equations.

Answer 2

Aegon's answer gives an idea of the meaning. I'd like to say a little bit about why you would want such a thing. The basic application is to solutions of differential equations, and the idea is the following abstract calculation. Suppose you want to be able to solve the differential equation

$$Lu=f$$

where $L$ is some differential operator, $u$ is the unknown function, and $f$ is a given function. We'll say it's on the full space to avoid boundary issues, although this is useful in boundary value problems as well.

When studying this, it would be nice if we had a single, unified approach, so that we could solve the equation for one $f$ and then get a representation of the solution for every other $f$. The way that we do this is based on your equation, which can be abstractly written as

$$\delta * f = f$$

where $*$ denotes convolution. This means that if we can find $g$ such that

$$Lg=\delta$$

then we can convolve with $f$ on both sides to get

$$Lg*f=f.$$

Finally if we can argue that $Lg*f=L(g*f)$, then we've solved the problem, and $g*f$ is our solution. This $g$ is called the Green's function or the fundamental solution for $L$. Like the Dirac delta, $g$ is never strictly speaking a function, it is always a distribution, although it is often a function away from $0$. It can be found explicitly in a number of very important examples, including linear ODEs with constant coefficients, and the three "classic" PDE, i.e. the Laplace, heat, and wave equations.