Distributions

Section 5.3 Distributions

In analysis (the world of transformation, non-elementary functions, etc.) I can extend the notion of a function in strange and novel ways. While I’m not going to give a formal definition, these extensions are called distributions. The basic idea is that a distribution may not be a well-defined function, but it is something that behaves well in integration. (If the word ‘distribution’ reminds you of probability and statistics, that’s a good intuition. These distributions are very similar to distributions used in statistics where, again, the ability to integrate is the most important property).

The distribution I want to share in this section is the \(\delta\)-function. (The name is terrible, since it is most certainly not a function.) Here is the definition. Since the most important thing to know about a distribution is how to integrate it, I’ll define the \(\delta\)-function by its integration property.

Definition 5.3.1.

The \(\delta\)-function centred at \(a \in \RR\) is the unique distribution that satisfies the following two integration properties. Let \(f(t)\) be any integrable function with domain \(\RR\text{.}\)

\begin{align*} \int_{-\infty}^\infty \delta_a(t) dt \amp = 1\\ \int_{-\infty}^\infty \delta_a (t) f(t) dt \amp = f(a) \end{align*}

This is a good definition, but not a very clear one. What is this distribution — how is it like a function and what function is it like? The best way to define the \(\delta\)-function is as a limit of other function, which I’ll try to do now.

Let \(b \in \NN\) and consider the bell curve functions \(\frac{\sqrt{b}}{\sqrt{\pi}} e^{-b(t-a)^2}\text{.}\) All of these functions have integral \(1\text{,}\) by design of the choice of the constnat involved.

\begin{equation*} \int_{-\infty}^\infty \frac{\sqrt{b}}{\sqrt{\pi}} e^{-b(t-a)^2} dt = \frac{\sqrt{b}}{\sqrt{\pi}} \int_{-\infty}^\infty e^{-b(t-a)^2} dt = \frac{\sqrt{b}}{\sqrt{\pi}} \frac{ \sqrt{\pi}}{\sqrt{b}} = 1 \end{equation*}

These functions are all bell curves, but they become taller and narrower as \(b\) increases. Figure 5.3.2 shows the progression of these bell curves.

Figure 5.3.2. Narrower and Narrower Bell Curves

Then I could define \(\delta_0(t)\) as the limit of the sequences of bell curves as \(b \rightarrow \infty\text{.}\)

\begin{equation*} \delta_0(t) = \lim_{b \rightarrow \infty} \frac{\sqrt{b}}{\sqrt{\pi}} e^{b(t-a)^2} \end{equation*}

The limit of a sequence of function is a strange idea and I’m not going to go into the technical details of limits of functions and the resulting distribution. (There is a theory for this, as part of real analysis.) But I can talk about the intuition. In the sequences of functions, the bell curves are becoming taller and narrower with each step. In the limit, I produce something that becomes infinitely narrow and infinitely tall. However, the area under the curve for any function in the limit is 1, so the value of the integral should be unchanged.

\begin{equation*} \int_{-\infty}^\infty \delta_0(t) dt = 1 \end{equation*}

Since the result of the limit is essentially just an infinitely tall, infinitely narrow spike, sometimes the \(\delta\)-function is defined as follows.

\begin{equation*} \delta_a(t) = \left\{ \begin{matrix} \infty \amp t = a \\ 0 \amp t \neq a \end{matrix} \right. \end{equation*}

This is very strange, since \(\infty\) is not a number and shouldn’t be the output of a function. Indeed, this is not a function (even though it is named, poorly, as ‘the \(\delta\)-function’). But this does get at the idea of the \(\delta\)-function. In many introductory situations, this piecewise definition is, indeed, given as the definition; I prefer to give the integration definition, since it is better practice.

The previous limit defined the \(\delta\)-function centred at \(a=0\text{.}\) In many books, this is simiply written \(\delta(t)\text{,}\) without the subscript. The other \(\delta\)-function are simply shifts of this distribution.

\begin{equation*} \delta_a(t) = \delta_0(t-a) \end{equation*}

Finally, I could integrate the product of the bell curve sequence and some other function \(f(t)\) with domain \(\RR\text{.}\) If I integrate \(f(t) \frac{\sqrt{a}}{\sqrt{\pi}} e^{-at^2}\text{,}\) I get a weighted average of \(f(t)\) values near \(a\text{.}\) In the limit, though, only the value at \(f(a)\) matters, since the \(\delta\)-function multiplies by zero everywhere else. The weighted average of just one value is simply that value, so I can argue (informally) for the second property of the \(\delta\)-function.

\begin{equation*} \int_{-\infty}^\infty \delta_a (t) f(t) dt = f(a) \end{equation*}

Since the \(\delta\)-function, and distributions in general, can be integrated, I can take their Laplace transforms. The Laplace transform for the \(\delta\)-function is very easy to calculate. (Indeed, the \(\delta\)-function is the nicest and easiest function to integrate in basically any circumstance due to its second property.)

\begin{equation*} \calL \{ \delta_a(t) \} (s) = \int_0^\infty \delta_a (t) e^{-st} dt = \left\{ \begin{matrix} 0 \amp a \lt 0 \\ 1 \amp a = 0 \\ e^{-as} \amp a \geq 0 \end{matrix} \right. \end{equation*}

This is quite an odd result. I started with a distribution which wasn’t even a proper function, but it’s Laplace transform is a proper, well-behaved differentiable function. Laplace transforms don’t even exclusively send functions to functions, but allow for distributions as well.

Before I end this section, I can ask why I would define such a strange function. Think about harmonic systems and forcing terms again. The \(\delta\)-function can act as a forcing term; if it does, it represents an instantaneous jolt to the system. The standard image of a harmonic system is a mass on a spring. In this image, a \(\delta\)-function represents hitting the mass with a hammer at one moment in time. The force only acts for an instant, but it transfers some finite energy and causes a change in the system. Like step functions modelling switches, the \(\delta\) functions models sudden impact, which is very reasonable and useful for applications.

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