This is hardly a proof, but I understand that if m is less than n by one it can be thought of as the du of a u substitution.
I'm interested in the other cases though...
A recent thread in the calculus forum reminded me of the following statement, which I've seen in a book:
If , then can be expressed in terms of elementary functions only when:
(a) is an integer, (b) is an integer, or (c) is an integer. My question: how is this proven?
My understanding is that people use Differential Galois theory to prove statements like this.
Differential Galois theory - Wikipedia, the free encyclopedia
http://nd.edu/~mkamensk/lectures/diffgalois.pdf
I just learnt today that this actually a theorem. It's called 'Chebyshev theorem' (so the three conditions are spoken of as 'Chebyshev conditions'). There's also something called 'Chebyshev Integral', which is a special case.