Associate 1987-88

Bruce C Berndt


Ramanujan's Notebooks

Srinivasa Ramanujan is generally acknowledged to be India's greatest mathematician. Born in 1887, he lived most of his life in obscurity and poverty while recording his mathematical discoveries in notebooks during the period of 1903-1914, approximately. In 1914, Ramanujan left India for England at the firm persuasion of England's famed mathematician G. H. Hardy, to whom Ramanujan had written in 1913 with statements of some of his theorems. In the next three years, Ramanujan received world-wide acclaim for his mathematics. However, in 1917, Ramanujan contracted an incurable disease. Having returned to India, he died in 1920 at the age of 32.

Professor Berndt began to edit Ramanujan's notebooks in May, 1977. If a formula or theorem is considered to be new, he attempts to prove it. If a formula or theorem is thought to be known, he finds sources for it. Thus, some of his work involves historical scholarship. However, by far, the great majority of his time is spent in trying to prove the hitherto unproven results found in the notebooks. During his CAS appointment, Professor Berndt will work on completing Chapters 20 and 21 from Ramanujan's second notebook. This material is on modular equations, and almost all of it is new. The extent of Ramanujan's work in this field is truly remarkable, and it is no exaggeration to assert that he found more modular equations than all of his predecessors combined. Moreover, his modular equations of a given order are usually simpler and more elegant than those of his forerunners. In the previous two years, Remanujan's modular equations have assumed increased importance because of their role in fast computer computation.

Some of the initial material in the 100 unorganized pages after chapter 21 constitute the beginning of a theory analogous to a classical theory of elliptic functions featuring the theta-functions. To be brief, Ramanujan starts to develop a theory in which the role of the hypergeometric function F(1/2, 1/2; 1;x) in the classical theory is replaced by F(1/3,2/3; 1;x). In correspondence, Professor Berndt has learned that such a theory would be of great importance in physics. Although Ramanujan published only one continued fraction in his short lifetime, his letter to Hardy and his notebooks contain many deep and elegant continued fraction formulas. Indeed, in expanding functions into continued fractions and in the converse problem of determining closed form expressions for continued fractions, Ramanujan is unequaled.