Ramanujan's lost notebook is the manuscript in which the Indian mathematician
Srinivasa Ramanujan
Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis ...
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
The Wren Library is the library of Trinity College in Cambridge. It was designed by Christopher Wren in 1676 and completed in 1695.
Description
The library is a single large room built over an open colonnade on the ground floor of Nevile' ...
at
Trinity College, Cambridge
Trinity College is a constituent college of the University of Cambridge. Founded in 1546 by Henry VIII, King Henry VIII, Trinity is one of the largest Cambridge colleges, with the largest financial endowment of any college at either Cambridge ...
. 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 (informally known as Madras University) is a public university, public State university (India), state university in Chennai, Tamil Nadu, India. Established in 1857, it is one of the oldest and among the most prestigiou ...
. 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
John Macnaghten Whittaker Fellow of the Royal Society, FRS FRSE LLD (7 March 1905 – 29 January 1984) was a British mathematician and Vice-Chancellor of the University of Sheffield from 1953 to 1965.
Life
Whittaker was born 7 March 1905 i ...
examined Watson's papers (which were a complete mess, due to be incinerated in a few days) and 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, 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.
The concept of the sabbatical is based on the Biblical practice of ''shmita'' (sabbatical year), which is related to agriculture. According to ...
, he wrote to
British
British may refer to:
Peoples, culture, and language
* British people, nationals or natives of the United Kingdom, British Overseas Territories, and Crown Dependencies.
** Britishness, the British identity and common culture
* British English, ...
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 (, , ; german: Straßburg ; gsw, label=Bas Rhin Alsatian, Strossburi , gsw, label=Haut Rhin Alsatian, Strossburig ) is the prefecture and largest city of the Grand Est region of eastern France and the official seat of the Eu ...
, near the
France
France (), officially the French Republic ( ), is a country primarily located in Western Europe. It also comprises of Overseas France, overseas regions and territories in the Americas and the Atlantic Ocean, Atlantic, Pacific Ocean, Pac ...
-
Germany
Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populous country in Europe after Russia, and the most populous member state of the European Union. Germany is situated betwe ...
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 mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
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 analyti ...
, 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, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
of
black hole
A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
s.
"Ramanujan’s Lost Notebook"
''ThatsMaths''
References
Bibliography
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* Reprinted 2008
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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