Dirac delta function

Description

Let $C_c^\infty(\mathbb{R})$ be the space of infinitely differentiable functions $\varphi : \mathbb{R} \to \mathbb{C}$ that vanish outside a bounded set. The Dirac delta is the distribution $\delta$ that maps each test function $\varphi \in C_c^\infty(\mathbb{R})$ to $\delta[\varphi] = \varphi(0)$. Informally, one writes $\int_{-\infty}^{\infty} \delta(t)\, \varphi(t)\, dt = \varphi(0)$, as if $\delta$ were a function that is $0$ away from $0$ and has total integral $1$. No actual function has this property. The delta is a generalized function.

Remarks

As a measure, $\delta$ gives the mass $1$ to each set that contains $0$ and the mass $0$ to the other sets. The shifted delta $\delta(t - a)$ maps $\varphi$ to $\varphi(a)$. For $a > 0$, its Laplace transform is $e^{-a s}$, and for $a = 0$ the usual convention gives $1$. In the sense of distributions, $\delta$ is the derivative of the Heaviside step function. In differential equations, $\delta(t - a)$ models an impulse: a large force that acts for a very short time around $t = a$.

Explanations

L1 human

The delta function is an idealized spike at $0$, infinitely tall and infinitely thin, with area $1$. It is not a real function. But it makes sense when we integrate it against a smooth function, and the result is the value of that function at $0$.