So the formula for an ellipse in polar coordinates is r(θ) = p/(1+εcos(θ)). By evaluating L = ∫r(θ) dθ on the complex plane on a circle of circumference ε on the centered at the origin I obtained the equation L = (2π)/√(1-ε^2). Why then does Wikipedia say that the formula for the perimeter is an elliptic integral? I thought we couldn't come up with a closed form solution to an elliptic integral? I also saw someplace on the internet that this is only an approximate solution? Is Cauchy's formula only an approximation?