A reader brought up essentially this question: does anyone happen to know a proof that does not possess an elementary antiderivative? By “elementary”, I mean a member of the class of functions which contains all constants valued in the complex numbers, the identity function, the exponential and log functions, and closed under the four basic arithmetic operations and composition.

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October 21, 2008 at 4:47 pm

Charles SiegelI don’t know the details offhand, but I’ve seen the proof using differential galois theory. I would think any book with a title like “Differential Algebra” would have it.

October 21, 2008 at 5:30 pm

SergeyAs far as i know it was first shown in Liouville in «Sur la determination des integrales dont la valeur est algebrique».

October 21, 2008 at 5:37 pm

SergeyYou can see the bibliographical reference this article in russian: http://www.mccme.ru/free-books/matpros/i8126135.pdf.zip

October 21, 2008 at 5:47 pm

Todd TrimbleThanks, guys. As I’ve learned through a quick search (now that I have a moment), there’s something called the Risch algorithm which apparently decides whether an elementary admits an elementary antiderivative. But I haven’t taken a closer look yet, and it’s not clear to me yet whether the general problem is actually decidable in a logical sense.

October 21, 2008 at 11:18 pm

Henry CohnIt’s not decidable (D. Richardson, Some unsolvable problems involving elementary functions of a real variable, J. Symbolic Logic 33 (1968), 514-520). See also the discussion on pages 175-176 of Risch’s paper.

Years ago Matthew Wiener posted to usenet a nice account of how one proves that certain functions don’t have antiderivatives. It’s available here: http://www.math.niu.edu/~rusin/known-math/97/nonelem_integr2

There’s also a good exposition in M. Rosenlicht, Integration in finite terms, Amer. Math. Monthly 79 (1972), 963-972.

October 22, 2008 at 10:04 pm

Todd TrimbleHenry, once again you provide some very useful information! The Wiener article in particular is well done and puts the matter to rest; I had found reference to the Rosenlicht article by poking around myself, but haven’t had a chance to go down to the library to take a look.

Thanks very much!

October 23, 2008 at 7:53 am

JustinThanks for posting this Todd!

November 13, 2008 at 2:32 am

Scott CarnahanBrian Conrad has an “explanation for talented high school students” on his web page (scroll down to the Notes section), although there is a minus sign in the exponent.

November 13, 2008 at 3:09 am

Todd TrimbleThanks, Scott. I just skimmed it (link here), and it looks like a good article; basically it shows how we can prove certain functions have no elementary antiderivative

ifwe assume Liouville’s criterion. (For the proof of Liouville’s criterion, he refers the reader to Rosenlicht’s article cited above.) He fleshes out a lot of what I found in slightly more abbreviated form in the Wiener article cited above, and adds some more interesting examples; the one I’d like to understand better is elliptic functions [example 4.3], where he addresses more advanced readers with an explanation which I didn’t quite follow.