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Ramanujan's lost notebook is the manuscript in which the Indian mathematician
Srinivasa Ramanujan Srinivasa Ramanujan Aiyangar (22 December 188726 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial con ...
recorded the mathematical discoveries of the last year (1919–1920) of his life. Its whereabouts were unknown to all but a few mathematicians until it was rediscovered by George Andrews in 1976, in a box of effects of G. N. Watson stored at the Wren Library at
Trinity College, Cambridge Trinity College is a Colleges of the University of Cambridge, constituent college of the University of Cambridge. Founded in 1546 by King Henry VIII, Trinity is one of the largest Cambridge colleges, with the largest financial endowment of any ...
. The "notebook" is not a book, but consists of loose and unordered sheets of paper described as "more than one hundred pages written on 138 sides in Ramanujan's distinctive handwriting. The sheets contained over six hundred mathematical formulas listed consecutively without proofs." have published several books in which they give proofs for Ramanujan's formulas included in the notebook. Berndt says of the notebook's discovery: "The discovery of this 'Lost Notebook' caused roughly as much stir in the mathematical world as the discovery of Beethoven’s tenth symphony would cause in the musical world."


History

After Ramanujan died on April 26, 1920, at the age of 32, his wife gave his notebooks to the
University of Madras The University of Madras is a public university, public State university (India), state university in Chennai (Madras), Tamil Nadu, India. Established in 1857, it is one of the oldest and most prominent universities in India, incorporated by an ...
. On August 30, 1923, the registrar Francis Drewsbury sent much of this material to
G. H. Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
, Ramanujan's mentor at Trinity College, where he probably received the manuscripts of the lost notebook. Some time between 1934 and 1947, Hardy probably passed the notebook on to G. N. Watson, who with B. M. Wilson started on the project of editing Ramanujan's notebooks. However, Wilson died in 1935 and Watson seems to have lost interest in the project in the late 1930s. After Watson's death in 1965, J. M. Whittaker examined Watson's papers (which were in disarray, covered a room floor to a depth of one foot, and due to be incinerated in a few days) and by luck found Ramanujan's notebook, which he and R. A. Rankin sent to Trinity College Wren library on December 26, 1968. , following a suggestion by Lucy Slater, found the lost notebook in the spring of 1976 while on a visit to Trinity College. It was published on December 22, 1987, by Narosa publishing house.


Andrews' account of the discovery

George Andrews, an American mathematician, wrote in 2012 an account of the discovery for the 125th celebration of Ramanujan's birth. In his account, Andrews states that he was already an advanced researcher in fields, such as mock theta functions and
hypergeometric series In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is ...
, related closely to works of Ramanujan. In 1970, anticipating a
sabbatical A sabbatical (from the Hebrew: (i.e., Sabbath); in Latin ; Greek: ) is a rest or break from work; "an extended period of time intentionally spent on something that’s not your routine job." The concept of the sabbatical is based on the Bi ...
, he wrote to
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mathematician Lucy Slater. Slater "intriguingly" stated in her reply that she had inherited a "great collection" of papers from mathematicians such as Watson, Bailey, Jackson and Rogers, which were unsorted, including one of the last by Ramanujan. She also mentioned other papers were held by the Trinity College library. Although unable to travel to Europe in 1970, Andrews became able to do so in 1976, when he was due to attend a European conference in
Strasbourg Strasbourg ( , ; ; ) is the Prefectures in France, prefecture and largest city of the Grand Est Regions of France, region of Geography of France, eastern France, in the historic region of Alsace. It is the prefecture of the Bas-Rhin Departmen ...
, near the Franco-German border. He obtained permission and support from Slater, from the Trinity College library, and from his professor, Ben Noble, to visit Cambridge after the conference, in order to investigate the "invaluable" unpublished writings of Watson ''et al''. Noble agreed, adding that if he could attempt to find a lost paper by
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish physicist and mathematician who was responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism an ...
at the same time, it would be appreciated. The library's documents included a list of matters held from Watson's estate. The list included the item: ''"A 139 page manuscript by S. Ramanujan on q-series"'', containing the work from Ramanujan's final year. Although not labelled as such, the identity of the papers was settled because Ramanujan's final letters to Hardy had referred to the discovery of what Ramanujan called mock theta functions, although without great detail, and the manuscript included what appeared to be his full notes on these.


Contents

described the lost notebook in detail. The majority of the formulas are about ''q''-series and mock theta functions, about a third are about modular equations and singular moduli, and the remaining formulas are mainly about integrals,
Dirichlet series In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in anal ...
, congruences, and asymptotics. The mock theta functions in the notebook have been found to be useful for calculating the
entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
of
black hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
s."Ramanujan’s Lost Notebook"
''ThatsMaths''


References


Bibliography

* * * * * * * * * * Reprinted 2008 *


Notes


External links


Andrews' paper on the finding of the Lost Notebook
in 1976 * * {{cite web , title=Ramanujan's Notebooks , first1=K. , last1=Srinivasa Rao , url= http://www.imsc.res.in/~rao/ramanujan/notebookindex.htm History of mathematics Mathematics literature Srinivasa Ramanujan