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Ramanujan's Notebooks: Part II
During the years 1903-1914, Ramanujan recorded many of his mathematical discoveries in notebooks without providing proofs. Although many of his results were already in the literature, more were not. Almost a decade after Ramanujan's death in 1920, G.N. Watson and B.M. Wilson began to edit his notebooks but never completed the task. A photostat edition, with no editing, was published by the Tata Institute of Fundamental Research in Bombay in 1957. This book is the second of four volumes devoted to the editing of Ramanujan's Notebooks. Part I, published in 1985, contains an account of Chapters 1-9 in the second notebook as well as a description of Ramanujan's quarterly reports. In this volume, we examine Chapters 10-15 in Ramanujan's second notebook. If a result is known, we provide references in the literature where proofs may be found; if a result is not known, we attempt to prove it. Not only are the results fascinating, but, for the most part, Ramanujan's methods remain a mystery. Much work still needs to be done. We hope readers will strive to discover Ramanujan's thoughts and further develop his beautiful ideas.
- GenresMathematics
372 pages, Hardcover
First published November 21, 1988
2 people are currently reading
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About the author
Srinivasa Ramanujan
22 books51 followersSrinivasa Ramanujan FRS (22 December 1887 – 26 April 1920) was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. Living in India with no access to the larger mathematical community, which was centred in Europe at the time, Ramanujan developed his own mathematical research in isolation. As a result, he rediscovered known theorems in addition to producing new work. Ramanujan was said to be a natural genius by the English mathematician G. H. Hardy, in the same league as mathematicians such as Leonhard Euler and Carl Friedrich Gauss.
Ramanujan was born at Erode, Madras Presidency (now Tamil Nadu) in a Tamil Brahmin family of Thenkalai Iyengar sect. His introduction to formal mathematics began at age 10. He demonstrated a natural ability, and was given books on advanced trigonometry written by S. L. Loney that he mastered by the age of 12; he even discovered theorems of his own, and rediscovered Euler's identity independently. He demonstrated unusual mathematical skills at school, winning accolades and awards. By 17, Ramanujan had conducted his own mathematical research on Bernoulli numbers and the Euler–Mascheroni constant.
Ramanujan received a scholarship to study at Government College in Kumbakonam, which was later rescinded when he failed his non-mathematical coursework. He joined another college to pursue independent mathematical research, working as a clerk in the Accountant-General's office at the Madras Port Trust Office to support himself. In 1912–1913, he sent samples of his theorems to three academics at the University of Cambridge. G. H. Hardy, recognizing the brilliance of his work, invited Ramanujan to visit and work with him at Cambridge. He became a Fellow of the Royal Society and a Fellow of Cambridge. Ramanujan died of illness, malnutrition, and possibly liver infection in 1920 at the age of 32.
During his short lifetime, Ramanujan independently compiled nearly 3900 results (mostly identities and equations). Nearly all his claims have now been proven correct, although a small number of these results were actually false and some were already known. He stated results that were both original and highly unconventional, such as the Ramanujan prime and the Ramanujan theta function, and these have inspired a vast amount of further research. The Ramanujan Journal, an international publication, was launched to publish work in all areas of mathematics influenced by his work.
In December 2011, in recognition of his contribution to mathematics, the Government of India declared that Ramanujan's birthday (22 December) should be celebrated every year as National Mathematics Day, and also declared 2012 the National Mathematics Year.
On his religious views, Ramanujan practices Hinduism. He credited his acumen to his family goddess, Mahalakshmi of Namakkal. He looked to her for inspiration in his work, and claimed to dream of blood drops that symbolised her male consort, Narasimha, after which he would receive visions of scrolls of complex mathematical content unfolding before his eyes.
Hardy argued that Ramanujan's religiousness had been romanticised by Westerners and overstated—in reference to his belief, not practice—by Indian biographers.
A Disappearing Number is a recent British stage production by the company Complicite that explores the relationship between Hardy and Ramanujan.
The novel The Indian Clerk by David Leavitt explores in fiction the events following Ramanujan's letter to Hardy.
On 22 March 1988, the PBS Series Nova aired a documentary about Ramanujan, "The Man Who Loved Numbers" (Season 15, Episode 19).
Ramanujan was born at Erode, Madras Presidency (now Tamil Nadu) in a Tamil Brahmin family of Thenkalai Iyengar sect. His introduction to formal mathematics began at age 10. He demonstrated a natural ability, and was given books on advanced trigonometry written by S. L. Loney that he mastered by the age of 12; he even discovered theorems of his own, and rediscovered Euler's identity independently. He demonstrated unusual mathematical skills at school, winning accolades and awards. By 17, Ramanujan had conducted his own mathematical research on Bernoulli numbers and the Euler–Mascheroni constant.
Ramanujan received a scholarship to study at Government College in Kumbakonam, which was later rescinded when he failed his non-mathematical coursework. He joined another college to pursue independent mathematical research, working as a clerk in the Accountant-General's office at the Madras Port Trust Office to support himself. In 1912–1913, he sent samples of his theorems to three academics at the University of Cambridge. G. H. Hardy, recognizing the brilliance of his work, invited Ramanujan to visit and work with him at Cambridge. He became a Fellow of the Royal Society and a Fellow of Cambridge. Ramanujan died of illness, malnutrition, and possibly liver infection in 1920 at the age of 32.
During his short lifetime, Ramanujan independently compiled nearly 3900 results (mostly identities and equations). Nearly all his claims have now been proven correct, although a small number of these results were actually false and some were already known. He stated results that were both original and highly unconventional, such as the Ramanujan prime and the Ramanujan theta function, and these have inspired a vast amount of further research. The Ramanujan Journal, an international publication, was launched to publish work in all areas of mathematics influenced by his work.
In December 2011, in recognition of his contribution to mathematics, the Government of India declared that Ramanujan's birthday (22 December) should be celebrated every year as National Mathematics Day, and also declared 2012 the National Mathematics Year.
On his religious views, Ramanujan practices Hinduism. He credited his acumen to his family goddess, Mahalakshmi of Namakkal. He looked to her for inspiration in his work, and claimed to dream of blood drops that symbolised her male consort, Narasimha, after which he would receive visions of scrolls of complex mathematical content unfolding before his eyes.
Hardy argued that Ramanujan's religiousness had been romanticised by Westerners and overstated—in reference to his belief, not practice—by Indian biographers.
A Disappearing Number is a recent British stage production by the company Complicite that explores the relationship between Hardy and Ramanujan.
The novel The Indian Clerk by David Leavitt explores in fiction the events following Ramanujan's letter to Hardy.
On 22 March 1988, the PBS Series Nova aired a documentary about Ramanujan, "The Man Who Loved Numbers" (Season 15, Episode 19).
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