Ramanujan’s Ellipse Formula and Radical Bounds

Abstract

Let P(a,b)be the perimeter of an ellipse with semiaxes a less then b > 0, and set h=(a-b/a+b)2. The normalized perimeter is the hypergeometric function F(h)=(a,b) pi (a+b)=2F1(-1 2,-1 2;1;h). We first characterize Ramanujan’s radical 3- radical symbol4-h as the largest global lower bound in a natural family of tangent-matched one-radical functions. We then introduce the quadratic-radical function S(h)= 2- radical symbol1- h2+h2 32. Itistheunique radicalwitha quadraticradicandthat agrees with F through h3. A coefficient argument proves 3- radical symbol4- hpunctuation markS(h)punctuation mark F(h) (0punctuation markhpunctuation mark1)and gives the best possible estimate 716384 h4 punctuation markF(h) - S(h)punctuation mark(4pi - 2+radical symbol17 32) h4. Thus, the new lower approximation has an eccentricity error of order e 16, compared with order e 12 for Ramanujan’s first formula. Finally, we give a canonical construction of higher-degree radical approximants and prove their local existence and uniqueness.