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As it is well-known, there is no formula for expression of perimeter of the ellipse $(\frac{x}{a})^2+(\frac{y}{b})^2=1$, as an elementary function of $a$ and $b$. I am interested to find an exact formulation of this fact, and its proof.

Especially, could one impose conditions on $a$ and $b$, such that an elementary function expression of perimeter is available? (Loosely speaking, similar to Galois theory which discusses about solvability of polynomial equations). Is there a purely algebraic formulation of the problem?

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    $\begingroup$ You should probably read a bit about elliptic integrals. $\endgroup$ Feb 14, 2017 at 21:43
  • $\begingroup$ I want a prove that shows that an elliptic integral is not in general expressible by means of elementary functions. Are there any algebraic proofs for example? @IttayWeiss $\endgroup$
    – XIII
    Jan 17, 2018 at 13:55
  • $\begingroup$ Chebyshev's theorem on elementary integrals states that $\int x^p(a+bx^r)^q\, dx$ has an elementary integral iff one of $(p+1)/r$, $q$, or $(p+1)/r + q.$ Now as far as I can see, the Legendre form of the elliptic integral of the second kind (which is the one that gives the perimeter of an ellipse) isn't quite of this form. It is $\int \frac{\sqrt{1-k^2x^2}}{\sqrt{1-x^2}}\,dx.$ But I think it is close enough that the same techniques work $\endgroup$
    – ziggurism
    Feb 26 at 23:38

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