Modular Equations in Ramanujan’s Lost Notebook

Abstract

Ramanujan recorded several hundred modular equations in his three notebooks [7]; no other mathematician has ever discovered nearly so many. Complete proofs for all the modular equations in Ramanujan’s three notebooks can be found in Berndt’s books [1]—[3]. In particular, Chapters 19—21 in Ramanujan’s second notebook are almost exclusively devoted to modular equations. Ramanujan used modular equations to evaluate class invariants, certain q-continued fractions including the Rogers-Ramanujan continued fraction, theta-functions, and certain other quotients and products of theta-functions and eta-functions [3].