On a generalization of Jacobi's elegantissima

Abstract

We establish a generalization of Jacobi's elegantissima, which solves the pendulum equation. This amazing formula appears in lectures by the famous cosmologist Georges Lema\^itre, during the academic years 1955-1956 and 1956-1957. Our approach uses the full power of Jacobi's elliptic functions, in particular imaginary time is crucial for obtaining the result.