At this rate I should change the name of this feature to Monday Function, but with any luck I can get my weekend schedule back to something approaching normal and break this streak of late SFs. But hey, every day is a good day for math.
This function is one we can define in a piece-by-piece way. It’s equal to zero everywhere except near the origin. Between -1/2 and 1/2, it’s just equal to 1. As such it’s just a rectangle hanging out at the middle of the number line.
The area of this rectangle is just its base times its height, or in this case 1 * 1 = 1. This isn’t the only way to get a rectangle of area 1, of course. We might shrink the base by a factor of two and raise the height by a factor of two. Then the base multiplied by the height would be 1/2 * 2 = 1. We could repeat the process – shrink the base, raise the height – and get a base of width 1/4 times a height of 4, again leaving a total area of 1. And we can imagine doing this over and over again until we have a microscopically thin and tremendously tall needle of a rectangle towering over the landscape.
The needle, eccentric though it is, still has an area of 1. If we decide to go ahead and give one of these needle functions a name – say, delta(x) – we can see if it has any interesting properties. Let’s take this delta function and multiply it by a different function. It doesn’t really matter what that different function is, so we’ll just call it f(x). What do we get?
Assuming the function f(x) isn’t too much of a screwball, it ought to be true that its value doesn’t change much over the width of that needle. And if that’s true then we have an approximate answer. The two functions multiplied together will themselves form a rectangle in the same location. The new height is the old height (call it h), times the value of the function f at that point. Put simply, the new height is h*f(0), because the rectangle is located at x = 0. But the width of the rectangle is 1/h by our defining area property, and so the total area is just the base (1/h) times the height h*f(0). Multiplying, we see that the area is just f(0).
In other words the delta function picks out the value of the other function at the location of the rectangle when you find their total area. If we put don’t require the delta function to be centered at the origin but instead put it at position a, we can say:
In physics and mathematics we’d prefer to drop that approximation and find something exact. To do this, we could say that the width of the rectagle is actually 0 and the height is infinity, in such a way as to produce a total area of 1. If this sounds suspiciously implausible, don’t worry. It is in fact totally impossible. Instead, some very clever people asked what kinds of mathematical objects could do this kind of job and the theory of distributions works just fine. The delta function isn’t strictly speaking a function anymore, but for our purposes we can still think of it as that infinitely thin spike. The actual object is called the Dirac delta function, and it’s defined in such a way so that multiplied by another function and integrated, it obeys the same sort of property as above:
This is monumentally useful, much more so that you might think at first. Because this identity is so simple, it turns out that the delta function can be used to easily represent many otherwise much more complicated processes. The impact of a bat on a baseball can be thought of as a delta function impulse, a sine wave can be thought of as a delta function in frequency space, you name it.
We are not, by the way, restricted to constructing a delta function representation with just these rectangles. There’s nearly endless other families of functions that can be used; Gaussian functions are probably the most common.
Not bad for something that’s not really a function at all.