I recently discovered there’s no closed-form formula for the circumference of an ellipse, and then was also told “But that just means there’s no nice formula in what we consider basic functions.” Do you have a sense / opinion on how explicit / arbitrary such elliptic formulae in fact are?
It depends on context. Is the exponential explicit? For the last 200 years, the answer is “hell yeah”. Exponential, logarithm and trigonometry (complex exponential) appear very often in life, and people can be expected to have a working knowledge of how to manipulate them. Expressing a solution in terms of exponentials is like meeting an old friend.
120 years ago, knowing elliptic integrals, their theory and how to manipulate them was considered basic knowledge that every working mathematician or engineer was expected to have. Back then, these were explicit / basic / closed form.
If you are writing a computer algebra system of similar ambition to maple / mathematica / wolfram alpha, then you better consider them explicit in your internal simplification routines, and write code for manipulating them. Otherwise, users will complain and send you feature requests. If you work as editor at the “Bronstein mathematical handbook”, then the answer is yes for the longer versions of the book, and a very hard judgement call for shorter editions.
Today, elliptic integrals are not routinely taught anymore. It is a tiny minority of mathematicians that has working knowledge on these guys. Expressing a solution in terms of elliptic integrals is not like meeting an old friend, it is like meeting a stranger who was famous a century ago, a grainy photo of whom you might have once seen in an old book.
I personally would not consider the circumference of an ellipse “closed form”. Just call it the “circumference of the ellipsis”, or write it as an integral, depending on how to better make apparent which properties you want.
Of course this is a trade-off, how much time to spend developing an intuition and working knowledge of “general integrals” (likely from a functional analysis perspective, as an operator) and how much time to spend understanding specific special integrals. The specific will always be more effective and impart deeper knowledge when dealing with the specifics, but the general theory is more applicable and “geometric”; you might say that it extrapolates very well from the training set. Some specific special functions are worth it, eg exp/log, and some used to be considered worthy but are today not considered worthy, evidenced by revealed preference (what do people put into course syllabi).
So, in some sense you have a large edifice of “forgotten knowledge” in mathematics. This knowledge is archived, of course, but the unbroken master-apprentice chains of transmission have mostly died out. I think this is sad; we, as a society, should be rich enough to sponsor a handful of people to keep this alive, even if I’d say “good riddance” for removing it from the “standard canon”.
Anecdote: Elliptic integrals sometimes appear in averaging: You have a differential equation (dynamical system) and want to average over fast oscillations in order to get an effective (ie leading order / approximate) system with reduced dimension and uniform time-scales. Now, what is your effective equation? You can express it as “the effective equation coming out of Theorem XYZ”, or write it down as an integral, which makes apparent both the procedure encoded in Theorem XYZ and an integral expression that is helpful for intuition and calculations. And sometimes, if you type it into Wolfram alpha, it transforms into some extremely lenghty expression containing elliptic integrals. Do you gain understanding from this? I certainly don’t, and decided not to use the explicit expressions when I met them in my research (99% of the time, mathematica is not helpful; the 1% pays for the trivial inconvenience of always trying whether there maybe is some amazing transformation that simplifies your problem).