Legendre’s Singular Modulus

Abstract

In 1811, Legendre published an identity between integrals involving the constant $k:=\hspace{0.17em}\text{sin}\hspace{0.17em}\frac{\pi }{12}$ that inspired Abel to create his brilliant theory of complex multiplication. Then k reappeared as the eccentricity of an ellipse whose arclength Ramanujan computed explicitly in terms of gamma functions with rational arguments. Finally, the constant appeared as a consequence of the three-body choreography along Bernoulli’s lemniscate. We develop these results in detail as well as mentioning random walks on a cubic lattice and the renormalization of the period of the simple pendulum. All this shows the wonderful unity underlying seemingly different branches of mathematics and physics.

ACKNOWLEDGMENT

We thank Carlo Beenakker for his help with the proof of (15), and also Joseph C. Várilly and Adrián Barquero for useful comments. We also thank a referee for his comments on the lemniscate choreography. Financial support from the Vicerectoría de Investigación of the University of Costa Rica is acknowledged.

Additional information

Funding

We thank Carlo Beenakker for his help with the proof of (15), and also Joseph C. Várilly and Adrián Barquero for useful comments. We also thank a referee for his comments on the lemniscate choreography. Financial support from the Vicerectoría de Investigación of the University of Costa Rica is acknowledged.

Notes on contributors

Mark B. Villarino

Mark B. Villarino has been a member of the School of Mathematics at the University of Costa Rica since 1974 and is currently an Associate Professor. His major interests are the theory of numbers, classical analysis, Ramanujan, and the history of mathematics.