Yesterday my friend Fiza asked me about an interesting mathematical problem. She stumbled across this problem while doing her research on cosmology. She wanted to compute the integral, $$ \int_0^x \frac{dx}{\sqrt{\cos x}} $$ As elementary as it may look, it turned out to be very cool mathematically.
Let's find out what's behind it. My first guess was that the integral was a fancy elliptic integral so I piked some books on the subject off of my shelf and looked at the formulae.
For those of you who are interested, here they are. The letters $\theta$ and $\alpha$ are called the modular angles (although I personally really think the name "elliptic angle" would have been better) and the letter $k$ is the Legendre modulus. It's called like that because it classifies the elliptic curves up to an anharmonic substitution (more on that in another post).
Incomplete elliptic integral of the fourth kind. Heu.. no wait, there are only three kinds. By the way, the term "incomplete integral" is the old fashioned way to call a primitive function.
At this stage, the integral that looks the most like Fiza's integral is the incomplete elliptic integral of the first kind. Except that the cosine is not squared. But that's not much of a problem, look, $$ \begin{aligned} \cos x &= \cos(\tfrac{1}{2}x +\tfrac{1}{2}x) \\ &= \cos^2(\tfrac{1}{2}x) - \sin^2(\tfrac{1}{2}x) \\ &= 1 - 2 \,\sin^2(\tfrac{1}{2}x) \end{aligned} $$ and voila, $$ \begin{aligned} \int_0^x \frac{1}{\sqrt{\cos x}}\, dx &=\int_0^{x} \frac{dx}{\sqrt{1 - 2\,\sin^2(\tfrac{1}{2}x)}} \\ &= 2 \int_0^{x/2} \frac{d\theta}{\sqrt{1 - 2\, \sin^2\theta}}\\ &= 2 \, F\left(\tfrac{1}{2} x, \sqrt{2} \right) \end{aligned} $$
This is funny, there's nothing more to it. Problem solved. You are going to say: wait, you've not solved the problem, you've just put a fancy name on the integral that's it. Well it's totally true and indeed the problem is solved, that's how special functions work. They are extensively studied functions, once you can reduce your problem to a known special function, then your problem is solved because you connected the unknown (your problem) the the very well known (special functions).
But what exactly are special functions ? Well, there is no formal definition. They simply are especially nice functions occurring frequently in famous problems of mathematics or physics. They usually bear the name of the scientist who studied them at length. Most of them occur as solutions to simple differential equations not solvable by elementary means or, what comes out to be the same, as some integrals not expressible by the mean of elementary functions.
Elliptic integrals show up, for example, when you try to measure the arc length of an ellipse, hence their name. They have been studied by Gauss, Legendre, Abel and Jacobi at the beginning of the nineteenth century and they are the starting point of a very rich and colorful subject, namely the Theory of Elliptic Curves.
It was the beautiful discovery by Abel, that elliptic functions are inverse to the incomplete elliptic integrals, that launched the subject.
It's a real pleasure to find smoeone who can think like that