Elliptic integrals: do they actually exist? On my very first question on this website (Perimeter of an ellipse solution and elliptic integrals.) I ended up asking for an expression for
$$\frac{4}{a}\int_{0}^a\sqrt{(b^2-a^2)x^2+a^4\over ({a^2-x^2})}dx $$
and was told it was an elliptic integral.
EDIT:
Thank you to Angina Seng for providing the wikipedia link. My question now changes to, is there a proof that they cannot be represented by elementary functions?
 A: Your title and your question are different.
The definite integral you've written down is the integral of a nice-enough function over a reasonable domain, and turns out to exist and be finite. It's a real number that exists, even though you may not be able to calculate it precisely with perfect precision (a little like $\pi$ or $\sqrt{2}$ in that sense).
On the other hand, if you replaced the upper limit on your integral with some number $c$ with $|c| \le |a|$, then you'd have an expression that gives a number for every possible such $c$, which we could call $F(c)$. And you might ask "is there a way to write down a formula for $F$ using things I already know?" The answer there is that except in very exceptional circumstances (e.g., $b = a$), there is no such formula.
This shouldn't come as too big a surprise, although it may be disappointing. After all, when you looked at
$$
L(x) = \int_1^x \frac1t ~ dt,
$$
you had the same experience: it's well-defined and finite for every $x > 0$, but there's no "formula" for it...so we gave it a new name, and called it $\ln(x)$.
Exactly the same sort of thing is going on here.
Recally that by adding log to your arsenal of functions, you could suddenly integrate lots more things. By adding the elliptic integral functions, you can again integrate even more things. And just like logarithms, tables of values of elliptic integrals have been written down, so that you can compute to fairly high precision, and with computers to help, we can actually estimate these to pretty much any precision needed.
To answer your changed question: yes, there are proofs that certain elliptic functions are not "elementary". Can I point you to one? Not offhand. I'd guess that the words "Groebner basis" tend to be involved in the proofs, but I've never actually read one, so I'm not certain.
