The legendary genius
RAMANUJAN Essays and Surveys: Bruce C. Brendt, Robert A. Rankin Editors; Published by Hindustan Book Agency (India), 17, UB, Jawahar Nagar, New Delhi-110016. Price not mentioned.
THIS, AN Indian edition of the book published earlier by the American Mathematical Society, presents a fascinating and thought-provoking account of Ramanujan's life, from his birth at Kumbakonam on December 22, 1887 till his demise on April 26, 1920 at the young age of 32. The story is well told with the scientific ferment and personalities of his time.
Since the Indian postage stamp appeared in 1962 commemorating Ramanujan's 75th birthday, several articles appeared on his life and work. But, for the first time, we have a thought-provoking and timely agenda in this book, which will be of interest to a variety of readers. Each of the eight parts, into which the book is divided, sets the stage for understanding "a mathematician so great that his name transcends jealousies, the one superlatively great mathematician whom India has produced in the last 1000 years." (Prof. Neville, University of Reading)
It offers insight into the influence on Ramanujan's development before he left for England. For example, the assistance that S. Narayana Iyer gave is narrated: working on problems with Ramanujan at night after his work at the Port Trust Office, publishing Ramanujan's discoveries in the theory of prime numbers, composing letters to Hardy and offering home to Ramanujan's parents after his death. In 1903, there came into his hands Carr's Synopsis of Mathematics, a book containing the enunciation of some 6000 theorems, mostly in geometry, for the most part without proofs. Geometry did not appeal to Ramanujan but in algebra and calculus he found himself in a magic world. In proving one formula, he discovered many others; be began the practice of compiling a notebook, the first one became famous afterwards.
It was in April 1914 that Ramanujan arrived in Cambridge, where his name became well known. Early in 1919, he returned to Madras, a tired and sick man of 31, nursed by his wife. There were many public receptions in his honour. The conversations with his wife, Janaki Ammal recorded in part III are very moving.
Parts VI to VIII are completely mathematical, delving into Ramanujan's manuscripts and notebooks. Some representative topics are listed: modular functions with various surprising symmetries, nested radicals, elementary identities involving radicals, formulas for finding equal sums of powers and polynomial identities.
The calculation of "pi" (the ratio of any circle's circumference to its diameter) has something of a benchmark in computation. Ramanujan's approach is now incorporated in computer algorithms yielding millions of digits of "pi", when he knew nothing of computer programming. Sophisticated algebraic manipulation software has allowed further exploration of the road Ramanujan travelled alone and unaided 90 years ago.
There are many wonderful formulas contained in his "Notebooks" that revolve around integrals, infinite series and continued fractions. Bruce Berndt is now completing the Herculean task of editing these.
It seems no person in the history of mathematics possessed the skills that Ramanujan had in determining continued fractions for various functions or finding closed form representation for continued fraction. It is high time that our students study Ramanujan's theorems in university courses, instead of merely extolling him as an enigmatic Indian genius.
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