We’ve done a lot of discussion of the concept of integrals of a function here on this blog. Their definitions and applications are so broad as to defy any one-sentence description, but one of the most basic is the idea of the area under a curve. More precisely it’s the signed area under the curve – that is, the area above the x-axis is counted as positive and the area under counts as negative. Stealing the image from Wikipedia, which communicates the concept well:
In some cases you might have a function that exhibits a particular kind of symmetry about the origin. On one side of the origin the graph is shaped a certain way, and on the other side the graph is shaped the exact same way except for the fact that it’s mirrored with respect to the x-axis. We’ve talked about this classification before, but to brush up our terminology we’ll note again that this type of function is called an odd function. (As in “the opposite of even”, not “a really weird function”.)
As a representative example of this kind of function, we might pick this little mutant Gaussian:
The thing about odd functions is that their symmetry (in fact we’d usually call it their antisymmetry) makes the integral very easy to calculate in certain circumstances. If you want all the area under the curve starting with x at negative infinity and going all the way to x at positive infinity, it’s pretty clear that the area on the left will exactly cancel the area on the right. The result will be a total signed area of 0. Symbolically we’d say that for an odd function f,
Are there exceptions? Well, yes and no. If everything is perfectly behaved in a Riemannian sense, it’s exception free. If not, for instance if the function fails to have a well-defined integral, there might be exceptions.
As an example, take a relative of the sinc function we’ve talked about over the last few Sundays. I’m not sure this one has a name, but it’s our Sunday Function:
It’s an odd function. Unfortunately it also blows up at the origin – the integral doesn’t converge to a finite value on either side taken along. Because of this we’re not guaranteed that any method we happen to use to calculate the area will in fact result in cancellation. Strictly speaking, stick a fork in us. We’re done. But mathematicians have been thinking about these problems for a long time. They’ve developed all kinds of subtle and brilliant tools for bringing some of these functions to heel. One of them is called the Cauchy principle value. The idea is that we ignore the problematic origin point of the function, and see if it cancels properly arbitrarily close to the origin. Symbolically, the principal value is the result you get is in this case the result of:
And because the function is well-behaved outside the origin, this somewhat large expression does cancel as we properly expect an odd function to do, and thus the Cauchy principle value of the integral is 0.
While I’m sure this doesn’t count as six impossible thing before breakfast, it’s pretty close to one impossible thing. Of course it’s not actually impossible, mathematics never quite lets you get away with that. But sometimes it seems pretty close to magic anyway.