Elliptic integrals: do they actually exist?

Question

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?

They exist well enough to have a long wikipedia page https://en.wikipedia.org/wiki/Elliptic_integral — Angina Seng, Aug 26 '20 at 11:30
Elliptic integrals cannot be expressed in terms of regular functions. Same with $Ci(x), Si(x), Ei(x)$, and so on. Proving that they cannot be written in terms of regular functions is probably quite hard. What can be useful is that each one of these non-standard functions have a bunch of properties, and for that I personally find wolframalpha documentation most helpful, e.g. for elliptic integrals of the second kind, see: https://functions.wolfram.com/EllipticIntegrals/EllipticE/ . You can click on the tabs at the bottom of the page (where it says "182 formulas"). — Adam Rubinson, Aug 26 '20 at 11:37
I would use that link only as a reference unless you're an expert (or want to become one) in elliptic integrals. — Adam Rubinson, Aug 26 '20 at 11:41
Once a question has answers, it starts to be rude to alter it substantially, because it makes the answers look stupid. No one will mind if you ask a new question, informed by the answers you've already gotten. [As a for-instance, you've made the first line of my answer look weird changing the title away from "do elliptic integrals exist?"] In the future, go ahead and ask a new question, and leave your old one around for others to learn from. — John Hughes, Aug 26 '20 at 11:44
@JohnHughes I'm sorry, that was insensitive of me, I'll change it back. — A-Level Student, Aug 26 '20 at 11:46
a comment because a link is not considered an Answer: the answer allegedly can be found here http://math.stanford.edu/~conrad/papers/elemint.pdf . I do not claim to have read it, but it looks quite readable (but non-trivial) [edit: this is for proving its not expressible in elementary functions] — Calvin Khor, Aug 26 '20 at 11:49
@A-levelStudent: I think it's something almost all of us have done early in our use of the site, and it doesn't bug me a lot in this case, but I can at least try to help you establish good habits early. I hope we'll see plenty more from you. — John Hughes, Aug 26 '20 at 11:55
Given the above, you can perhaps ask a new question to explain §4-§5 of M. Rosenlicht, Integration in finite terms, American Math. Monthly 79 (1972), 963–972. This and example 4.3 of the above PDF should give the proof of the result. Of course, read the PDF first before asking, or you will not really know what you're asking for (and look like you're just asking for a lot of work without putting in any yourself). I found this paper by googling but I'm not sure if it is legal so I will not link it — Calvin Khor, Aug 26 '20 at 11:58
Do you want the taylor series expansion (approximation) of the elliptic integral of the second kind around $x=0$? — Adam Rubinson, Aug 26 '20 at 11:58
@AdamRubinson why can't we have a general series expansion for all $x$ values? — A-Level Student, Aug 26 '20 at 12:06
That’s a good question. “For an infinitely differentiable function $ f$, The Taylor series of $f$ need not converge (point-wise) at any point other than $ x=0$”. See: https://math.stackexchange.com/questions/12862/on-what-interval-does-a-taylor-series-approximate-or-equal-its-function for more info. That’s doesn’t imply you cannot have a “general” Taylor Series expansion (around $x=0$, say) for the elliptic integral of the second kind, where by “general” you mean “A taylor series that converges for all $x$”.. And for that, I don’t know. — Adam Rubinson, Aug 26 '20 at 12:31

score 3 · Accepted Answer · answered Aug 26 '20 at 11:36

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.

Ok, thanks for the detailed answer. Is it possible to approximate the elliptic integral by writing the integrand as a Taylor series and integrating that? — A-Level Student, Aug 26 '20 at 11:39
That's one of many ways to approximate functions. If you write a Taylor series around $x = d$, it's generally useful for estimating the value near $d$, but not good for $x$ far from $d$ (and that also applies to the integrated version, which is itself a Taylor series for the antidervative, generally speaking for nice functions). — John Hughes, Aug 26 '20 at 11:42
I see. Has the Maclaurin/Taylor series been calculated for this particular integrand? — A-Level Student, Aug 26 '20 at 11:43
See: https://www.wolframalpha.com/input/?i=E%28x%29 and / or https://functions.wolfram.com/EllipticIntegrals/EllipticE/06/01/ — Adam Rubinson, Aug 26 '20 at 11:45
Also: one question per question. If you're confused by an answer, ask about it. If you want to ask another question, go ahead and ask another question; that new question will then be easily searchable by others. — John Hughes, Aug 26 '20 at 11:46

Elliptic integrals: do they actually exist?

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