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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 '17 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 $\endgroup$
    – XIE
    Jan 17 '18 at 13:55

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