## DyingLoveGrape.## Differential Equations: The Laplace Transform, Part 2: Two Other Importan Functions.## Some more neat functions.There's two other main types of functions that work well with Laplace transform: the Dirac $\delta$ function and the heaviside function. Let's talk about the $\delta$ function first. ## Dirac $\delta$ "Function".Strictly speaking, the $\delta$ function is not a "function", but it's a nice tool to use with functions. The idea is to look at a limit of "normal curves" like this: Notice that these functions get thinner and thinner, steeper and steeper, but
The idea behind the Dirac function is that it stands for an "impulse" at some point; it is almost like a brief, strong jolt. It's a bit like the feeling right when you wake up from a terrible dream in a cold sweat, except in function form.
What can we do with $\delta_{a}(x)$? Well, since this post is about Laplace transformations, maybe we ought to try that. By definition,
\[{\mathcal L}(\delta_{a})(s) = \int_{-\infty}^{\infty} \delta_{a}(t)e^{-st}\,dt\]
But, note, $\delta_{a}(t)e^{-st} = 0$ for every $x\neq a$. The integral [Small Note: sometimes we make the $\delta$ function slightly "more" of a function by defining the point where it is infinitely tall to be 1 instead of $\infty$. We still have the property that the total integral is equal to 1, but this makes the $\delta$ function at least plot-able.] ## The Heaviside Function.This function was constructed by Oliver Heaviside, but I like to think that it's called the Heaviside function because it's "heavy" on one side. You'll see what I mean.
The Heaviside function looks pretty boring, but it turns out to be quite useful. Here's a picture of the Heaviside function $H_{0}(x) Most of the time when we multiply something by the Heaviside function it just sends a lot of the function to 0 and then the rest of it stays the same. For example, with the function $f(x) = e^{-x}\sin(2x)$ and the Heaviside function $H_{0}(x)$ we have their product is: The dotted part of the graph is the rest of $e^{-x}\sin(2x)$ that the heaviside function "sent to 0." The Heaviside function is nice if you want to cut off useless data; if, for example, you make $\$4$ an hour as a graduate student, then if you plot that function you don't want it to extend to the negative axis since there's no such thing as a negative hour. Let's look at what happens when we take the Laplace transform of the Heaviside function: \[\begin{align*}{\mathcal L}(H_{a})(s) &= \int_{-\infty}^{\infty} H_{a}[x]e^{-st}\,dt\\ &= \int_{a}^{\infty}e^{-st}\,dt\\ &= \frac{-1}{s}\left[e^{-st}\right]_{a}^{\infty}\\ &= \frac{-1}{s}(0 - e^{-sa}) = \frac{e^{-sa}}{s}\end{align*}\] ## To sum up,...Let's jot all of this down in a box, just so we can refer back to it.
## Next time...?There's a number of different ways to combine these kinds of functions, but generally speaking there isn't a nice way, given $f,g$ functions, to find ${\mathcal L}(f(t)g(t))(s)$ in terms of ${\mathcal L}(f)(s)$ and ${\mathcal L}(g)(s)$; one would expect the Laplace transform of the product to be the product of the Laplace transforms of each function, but this isn't true — but, if we make one slightly alteration and take something called the |