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 Lemaître, 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.