Why do we only have an approximation for every circumference for ellipse, but we cannot define a special ratio formula for each ellipse? Is it possible for people to use a computer to find the exact "infinite series" relationship between the circumference of the ellipse and the major axis and minor axis?
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This is the complete elliptic integral of the second kind. Series are well known, and many numerical analysis packages or languages like Mathematica provide them as a function call.
Additionally, the algorithm for computing the circumference of an ellipse (based on the arithmetic-geometric mean) isn't too long if your environment doesn't support computing the complete elliptic integral of the second kind, $E(m)$:
(Yes, I'm using Mathematica. Yes, I know Mathematica has