A125707010427

Ramanujan’s Ellipse Formula and Radical Bounds
K. Srinivasa Raghava¹, R. Sivaraman²

¹K. Srinivasa Raghava, Research Associate, Pie Mathematics Association, Choolaimedu, Chennai (Tamil Nadu), India.

²Dr. R. Sivaraman, Associate Professor, Department of Mathematics, Dwaraka Doss Goverdhan Doss Vaishnav College, Arumbakkam, Chennai (Tamil Nadu), India.  

Manuscript received on 06 September 2026 | First Revised Manuscript received on 15 September 2026 | Second Revised Manuscript received on 29 September 2026 | Manuscript Accepted on 15 October 2026 | Manuscript published on 30 October 2026 | PP: 28-32 | Volume-6 Issue-2, October 2026 | Retrieval Number: 100.1/ijam.A125707010427 | DOI: 10.54105/ijam.A1257.07010427

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Abstract: Let 𝑷(𝒂,𝒃) be the perimeter of an ellipse with semiaxes 𝒂 ≥ 𝒃 > 𝟎, and set 𝒉 = ( 𝒂−𝒃 𝒂+𝒃 ) 𝟐 . The normalized perimeter is the hypergeometric function 𝑭(𝒉) = 𝑷(𝒂,𝒃) 𝝅(𝒂+𝒃) = 𝟐𝑭𝟏 (− 𝟏 𝟐 , − 𝟏 𝟐 ; 𝟏;𝒉). We first characterize Ramanujan’s radical 𝟑 − √𝟒 − 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 − √𝟏 − 𝒉 𝟐 + 𝒉𝟐 𝟑𝟐. It is the unique radical with a Quadratstich and that agrees with 𝑭 through 𝒉 𝟑 .A coefficient argument proves 𝟑 − √𝟒 − 𝒉 < 𝑺(𝒉) < 𝑭(𝒉) (𝟎 < 𝒉 < 𝟏) and gives the best possible estimate 𝟕 𝟏𝟔𝟑𝟖𝟒 𝒉 𝟒 < 𝑭(𝒉) − 𝑺(𝒉) < ( 𝟒 𝝅 − 𝟐 + √ 𝟏𝟕 𝟑𝟐) 𝒉 𝟒 . 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.

Keywords: Ellipse Perimeter, Ramanujan Approximation, Hypergeometric Function, Algebraic Approximation, Sharp Inequality.
Scope of the Article: Number Theory