And now, two quick notes before we get to business:
1. God help me, but I’ve joined the Twitter bandwagon. Here I am, @BuiltOnFacts. Though it goes under this blog’s name, it is more of a “personal” account. So you’ll be reading some incomprehensible personal minutiae, random observations, wild assertions, and somewhat more politics that I typically introduce here. But you may enjoy it nonetheless, and if you ever have physics questions that can be answered in 140 characters feel free to fire away.
2. Speaking of politics, this paragraph is political and I dislike it when people shoehorn politics into non-political posts. As such I will not even be slightly offended in you skip this paragraph. Onward: I’m a little horrified at some of the the horror that’s followed Citizens United v. FEC. Some of it’s due to misreporting (corporate campaign contribution limits were upheld, for instance). Some of it’s rabbit-chasing with the “corporate personhood” debate, which actually has pretty much nothing to do with the case, which should have come out the same even if corporations weren’t “persons” (the 1A doesn’t confer rights on people (except for assembly), it prohibits the government from making laws that abridge certain things). Finally, pretty much all speech that rises above personal pamphleteering is corporate speech in some sense – including the speech you’re reading now. The decision is certified ACLU-kosher, and I highly recommend constitutional law professor Ilya Somin’s excellent look at the issue. You can certainly be a proud liberal and still cheer the decision.
Ok, how about an actual function? I’m picking out a function that contains oceans of depth, but we’re just going to dip our toe into the water to see what the water feels like. You may be familiar with the Legendre polynomials, which we’ve talked about on a few occasions. They’re just a set of ordinary everyday polynomials that happen to have certain useful properties. They’re numbered by the order of the polynomial. The zeroth order polynomial is 1, the first order is x, the second is .5(3x^2 – 1), well the coefficients can get a little complicated but they are just regular polynomials. Let me plot the first four on the same graph:
You can see that some of them are even (they’re symmetric about the y-axis) and some are odd (inverted with respect to the other side of the y-axis). We’ve explored this property in the past, too. More on this in a second.
Our function contains all of the Legendre polynomials in one fell swoop. It’s the generating function of the Legendre polynomials:
The Legendre polynomials are defined as the functions Pn(x) that make that equation true. Every single one of the properties of the Legendre polynomials from the recursion relationships to the differential equation to their orthogonality can be derived directly from the generating function. It can get somewhat complicated, so we’ll only derive one of the easiest properties today. Let’s derive the parity (even/odd) characteristics of the Legendre polynomials.
First, x and t are variables so we can let them be whatever we want. Let’s replace t with -t and x with -x:
Negative times negative is positive, so actually the middle expression is the same both here and in the original expression. Equating equals with equals, this means:
We can pull the -1 out from the parentheses:
Now here’s the key thing. This is a sum so we can’t just cancel the t willy-nilly. But we can recognize that each t^n is attached to a unique Pn and effectively serves just as a label. In other words, what’s true for the sum is also true term-by-term. If you’re skeptical, just spend a few moments thinking about it, or write down a few terms explicitly to see how it works. This means:
Now we can cancel the t^n:
Which is precisely the statement of alternating even/odd parity that we expected from the graph. Ok, I admit this one was a little esoteric. But I think it’s still pretty cool!