Srinivasa Ramanujan
Srinivasa Ramanujan | |
---|---|
Died | 26 April 1920 | (aged 32)
Citizenship | British Indian |
Education | |
Known for | |
Awards | Fellow of the Royal Society (1918) |
Scientific career | |
Fields | Mathematics |
Institutions | University of Cambridge |
Thesis | Highly Composite Numbers (1916) |
Academic advisors | |
Signature | |
Srinivasa Ramanujan[a] (22 December 1887 – 26 April 1920) was an Indian
Ramanujan initially developed his own mathematical research in isolation. According to
During his short life, Ramanujan independently compiled nearly 3,900 results (mostly
In 1919, ill health—now believed to have been hepatic amoebiasis (a complication from episodes of dysentery many years previously)—compelled Ramanujan's return to India, where he died in 1920 at the age of 32. His last letters to Hardy, written in January 1920, show that he was still continuing to produce new mathematical ideas and theorems. His "lost notebook", containing discoveries from the last year of his life, caused great excitement among mathematicians when it was rediscovered in 1976.
Early life
Ramanujan (literally, "younger brother of Rama", a Hindu deity)[12] was born on 22 December 1887 into a Tamil Brahmin Iyengar family in Erode, in present-day Tamil Nadu.[13] His father, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district, worked as a clerk in a sari shop.[14][2] His mother, Komalatammal, was a housewife and sang at a local temple.[15] They lived in a small traditional home on Sarangapani Sannidhi Street in the town of Kumbakonam.[16] The family home is now a museum. When Ramanujan was a year and a half old, his mother gave birth to a son, Sadagopan, who died less than three months later. In December 1889, Ramanujan contracted smallpox, but recovered, unlike the 4,000 others who died in a bad year in the Thanjavur district around this time. He moved with his mother to her parents' house in Kanchipuram, near Madras (now Chennai). His mother gave birth to two more children, in 1891 and 1894, both of whom died before their first birthdays.[12]
On 1 October 1892, Ramanujan was enrolled at the local school.[17] After his maternal grandfather lost his job as a court official in Kanchipuram,[18] Ramanujan and his mother moved back to Kumbakonam, and he was enrolled in Kangayan Primary School.[19] When his paternal grandfather died, he was sent back to his maternal grandparents, then living in Madras. He did not like school in Madras, and tried to avoid attending. His family enlisted a local constable to make sure he attended school. Within six months, Ramanujan was back in Kumbakonam.[19]
Since Ramanujan's father was at work most of the day, his mother took care of the boy, and they had a close relationship. From her, he learned about tradition and puranas, to sing religious songs, to attend pujas at the temple, and to maintain particular eating habits—all part of Brahmin culture.[20] At Kangayan Primary School, Ramanujan performed well. Just before turning 10, in November 1897, he passed his primary examinations in English, Tamil, geography, and arithmetic with the best scores in the district.[21] That year, Ramanujan entered Town Higher Secondary School, where he encountered formal mathematics for the first time.[21]
A
In 1903, when he was 16, Ramanujan obtained from a friend a library copy of
When he graduated from Town Higher Secondary School in 1904, Ramanujan was awarded the K. Ranganatha Rao prize for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer introduced Ramanujan as an outstanding student who deserved scores higher than the maximum.[30] He received a scholarship to study at Government Arts College, Kumbakonam,[31][32] but was so intent on mathematics that he could not focus on any other subjects and failed most of them, losing his scholarship in the process.[33] In August 1905, Ramanujan ran away from home, heading towards Visakhapatnam, and stayed in Rajahmundry[34] for about a month.[33] He later enrolled at Pachaiyappa's College in Madras. There, he passed in mathematics, choosing only to attempt questions that appealed to him and leaving the rest unanswered, but performed poorly in other subjects, such as English, physiology, and Sanskrit.[35] Ramanujan failed his Fellow of Arts exam in December 1906 and again a year later. Without an FA degree, he left college and continued to pursue independent research in mathematics, living in extreme poverty and often on the brink of starvation.[36]
In 1910, after a meeting between the 23-year-old Ramanujan and the founder of the Indian Mathematical Society, V. Ramaswamy Aiyer, Ramanujan began to get recognition in Madras's mathematical circles, leading to his inclusion as a researcher at the University of Madras.[37]
Adulthood in India
On 14 July 1909, Ramanujan married Janaki (Janakiammal; 21 March 1899 – 13 April 1994),[38] a girl his mother had selected for him a year earlier and who was ten years old when they married.[39][40][41] It was not unusual then for marriages to be arranged with girls at a young age. Janaki was from Rajendram, a village close to Marudur (Karur district) Railway Station. Ramanujan's father did not participate in the marriage ceremony.[42] As was common at that time, Janaki continued to stay at her maternal home for three years after marriage, until she reached puberty. In 1912, she and Ramanujan's mother joined Ramanujan in Madras.[43]
After the marriage, Ramanujan developed a hydrocele testis.[44] The condition could be treated with a routine surgical operation that would release the blocked fluid in the scrotal sac, but his family could not afford the operation. In January 1910, a doctor volunteered to do the surgery at no cost.[45]
After his successful surgery, Ramanujan searched for a job. He stayed at a friend's house while he went from door to door around Madras looking for a clerical position. To make money, he tutored students at Presidency College who were preparing for their Fellow of Arts exam.[46]
In late 1910, Ramanujan was sick again. He feared for his health, and told his friend R. Radakrishna Iyer to "hand [his notebooks] over to Professor Singaravelu Mudaliar [the mathematics professor at Pachaiyappa's College] or to the British professor Edward B. Ross, of the
Pursuit of career in mathematics
In 1910, Ramanujan met deputy collector V. Ramaswamy Aiyer, who founded the Indian Mathematical Society.[52] Wishing for a job at the revenue department where Aiyer worked, Ramanujan showed him his mathematics notebooks. As Aiyer later recalled:
I was struck by the extraordinary mathematical results contained in [the notebooks]. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department.[53]
Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras. One of the first problems he posed in the journal[30] was to find the value of:
He waited for a solution to be offered in three issues, over six months, but failed to receive any. At the end, Ramanujan supplied an incomplete[59] solution to the problem himself. On page 105 of his first notebook, he formulated an equation that could be used to solve the infinitely nested radicals problem.
Using this equation, the answer to the question posed in the Journal was simply 3, obtained by setting x = 2, n = 1, and a = 0.[60] Ramanujan wrote his first formal paper for the Journal on the properties of Bernoulli numbers. One property he discovered was that the denominators of the fractions of Bernoulli numbers (sequence A027642 in the OEIS) are always divisible by six. He also devised a method of calculating Bn based on previous Bernoulli numbers. One of these methods follows:
It will be observed that if n is even but not equal to zero,
In his 17-page paper "Some Properties of Bernoulli's Numbers" (1911), Ramanujan gave three proofs, two corollaries and three conjectures.[61] His writing initially had many flaws. As Journal editor M. T. Narayana Iyengar noted:
Mr. Ramanujan's methods were so terse and novel and his presentation so lacking in clearness and precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly follow him.[62] Ramanujan later wrote another paper and also continued to provide problems in the Journal. In a letter dated 9 February 1912, Ramanujan wrote:
Sir, Attached to his application was a recommendation from E. W. Middlemast, a mathematics professor at the Presidency College, who wrote that Ramanujan was "a young man of quite exceptional capacity in Mathematics".[66] Three weeks after he applied, on 1 March, Ramanujan learned that he had been accepted as a Class III, Grade IV accounting clerk, making 30 rupees per month.[67] At his office, Ramanujan easily and quickly completed the work he was given and spent his spare time doing mathematical research. Ramanujan's boss, Sir Francis Spring, and S. Narayana Iyer, a colleague who was also treasurer of the Indian Mathematical Society, encouraged Ramanujan in his mathematical pursuits.[68]
In the spring of 1913, Narayana Iyer, Ramachandra Rao and E. W. Middlemast tried to present Ramanujan's work to British mathematicians. M. J. M. Hill of University College London commented that Ramanujan's papers were riddled with holes.[69] He said that although Ramanujan had "a taste for mathematics, and some ability", he lacked the necessary educational background and foundation to be accepted by mathematicians.[70] Although Hill did not offer to take Ramanujan on as a student, he gave thorough and serious professional advice on his work. With the help of friends, Ramanujan drafted letters to leading mathematicians at Cambridge University.[71]
The first two professors, H. F. Baker and E. W. Hobson, returned Ramanujan's papers without comment.[72] On 16 January 1913, Ramanujan wrote to G. H. Hardy.[73] Coming from an unknown mathematician, the nine pages of mathematics made Hardy initially view Ramanujan's manuscripts as a possible fraud.[74] Hardy recognised some of Ramanujan's formulae but others "seemed scarcely possible to believe".[75]: 494 One of the theorems Hardy found amazing was on the bottom of page three (valid for 0 < a < b + 1/2):
Hardy was also impressed by some of Ramanujan's other work relating to infinite series:
The first result had already been determined by
I understand there is a clerkship vacant in your office, and I beg to apply for the same. I have passed the Matriculation Examination and studied up to the F.A. but was prevented from pursuing my studies further owing to several untoward circumstances. I have, however, been devoting all my time to Mathematics and developing the subject. I can say I am quite confident I can do justice to my work if I am appointed to the post. I therefore beg to request that you will be good enough to confer the appointment on me.[65]
Contacting British mathematicians
On 8 February 1913, Hardy wrote Ramanujan a letter expressing interest in his work, adding that it was "essential that I should see proofs of some of your assertions".[78] Before his letter arrived in Madras during the third week of February, Hardy contacted the Indian Office to plan for Ramanujan's trip to Cambridge. Secretary Arthur Davies of the Advisory Committee for Indian Students met with Ramanujan to discuss the overseas trip.[79] In accordance with his Brahmin upbringing, Ramanujan refused to leave his country to "go to a foreign land".[80] Meanwhile, he sent Hardy a letter packed with theorems, writing, "I have found a friend in you who views my labour sympathetically."[81]
To supplement Hardy's endorsement, Gilbert Walker, a former mathematical lecturer at Trinity College, Cambridge, looked at Ramanujan's work and expressed amazement, urging the young man to spend time at Cambridge.[82] As a result of Walker's endorsement, B. Hanumantha Rao, a mathematics professor at an engineering college, invited Ramanujan's colleague Narayana Iyer to a meeting of the Board of Studies in Mathematics to discuss "what we can do for S. Ramanujan".[83] The board agreed to grant Ramanujan a monthly research scholarship of 75 rupees for the next two years at the University of Madras.[84]
While he was engaged as a research student, Ramanujan continued to submit papers to the Journal of the Indian Mathematical Society. In one instance, Iyer submitted some of Ramanujan's theorems on summation of series to the journal, adding, "The following theorem is due to S. Ramanujan, the mathematics student of Madras University." Later in November, British Professor Edward B. Ross of Madras Christian College, whom Ramanujan had met a few years before, stormed into his class one day with his eyes glowing, asking his students, "Does Ramanujan know Polish?" The reason was that in one paper, Ramanujan had anticipated the work of a Polish mathematician whose paper had just arrived in the day's mail.[85] In his quarterly papers, Ramanujan drew up theorems to make definite integrals more easily solvable. Working off Giuliano Frullani's 1821 integral theorem, Ramanujan formulated generalisations that could be made to evaluate formerly unyielding integrals.[86]
Hardy's correspondence with Ramanujan soured after Ramanujan refused to come to England. Hardy enlisted a colleague lecturing in Madras, E. H. Neville, to mentor and bring Ramanujan to England.
Life in England
Ramanujan departed from Madras aboard the S.S. Nevasa on 17 March 1914.[90] When he disembarked in London on 14 April, Neville was waiting for him with a car. Four days later, Neville took him to his house on Chesterton Road in Cambridge. Ramanujan immediately began his work with Littlewood and Hardy. After six weeks, Ramanujan moved out of Neville's house and took up residence on Whewell's Court, a five-minute walk from Hardy's room.[91]
Ramanujan spent nearly five years in
Ramanujan was awarded a Bachelor of Arts by Research degree
On 6 December 1917, Ramanujan was elected to the London Mathematical Society. On 2 May 1918, he was elected a Fellow of the Royal Society,[98] the second Indian admitted, after Ardaseer Cursetjee in 1841. At age 31, Ramanujan was one of the youngest Fellows in the Royal Society's history. He was elected "for his investigation in elliptic functions and the Theory of Numbers." On 13 October 1918, he was the first Indian to be elected a Fellow of Trinity College, Cambridge.[99]
Illness and death
Ramanujan had numerous health problems throughout his life. His health worsened in England; possibly he was also less resilient due to the difficulty of keeping to the strict dietary requirements of his religion there and because of wartime rationing in 1914–18. He was diagnosed with tuberculosis and a severe vitamin deficiency, and confined to a sanatorium. In 1919, he returned to Kumbakonam, Madras Presidency, and in 1920 he died at the age of 32. After his death, his brother Tirunarayanan compiled Ramanujan's remaining handwritten notes, consisting of formulae on singular moduli, hypergeometric series and continued fractions.[43]
Ramanujan's widow,
A 1994 analysis of Ramanujan's medical records and symptoms by D. A. B. Young
Personality and spiritual life
While asleep, I had an unusual experience. There was a red screen formed by flowing blood, as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing.
—Srinivasa Ramanujan[104]
Ramanujan has been described as a person of a somewhat shy and quiet disposition, a dignified man with pleasant manners.
Hardy cites Ramanujan as remarking that all religions seemed equally true to him.
Similarly, in an interview with Frontline, Berndt said, "Many people falsely promulgate mystical powers to Ramanujan's mathematical thinking. It is not true. He has meticulously recorded every result in his three notebooks," further speculating that Ramanujan worked out intermediate results on slate that he could not afford the paper to record more permanently.[8]
Mathematical achievements
In mathematics, there is a distinction between insight and formulating or working through a proof. Ramanujan proposed an abundance of formulae that could be investigated later in depth. G. H. Hardy said that Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets the eye. As a byproduct of his work, new directions of research were opened up. Examples of the most intriguing of these formulae include infinite series for π, one of which is given below:
This result is based on the negative fundamental discriminant d = −4 × 58 = −232 with class number h(d) = 2. Further, 26390 = 5 × 7 × 13 × 58 and 16 × 9801 = 3962, which is related to the fact that
This might be compared to Heegner numbers, which have class number 1 and yield similar formulae.
Ramanujan's series for π converges extraordinarily rapidly and forms the basis of some of the fastest algorithms currently used to calculate π. Truncating the sum to the first term also gives the approximation 9801√2/4412 for π, which is correct to six decimal places; truncating it to the first two terms gives a value correct to 14 decimal places .
One of Ramanujan's remarkable capabilities was the rapid solution of problems, illustrated by the following anecdote about an incident in which
Imagine that you are on a street with houses marked 1 through n. There is a house in between (x) such that the sum of the house numbers to the left of it equals the sum of the house numbers to its right. If n is between 50 and 500, what are n and x?' This is a bivariate problem with multiple solutions. Ramanujan thought about it and gave the answer with a twist: He gave a continued fraction. The unusual part was that it was the solution to the whole class of problems. Mahalanobis was astounded and asked how he did it. 'It is simple. The minute I heard the problem, I knew that the answer was a continued fraction. Which continued fraction, I asked myself. Then the answer came to my mind', Ramanujan replied."[112][113]
His intuition also led him to derive some previously unknown identities, such as
for all θ such that and , where Γ(z) is the
In 1918, Hardy and Ramanujan studied the
In the last year of his life, Ramanujan discovered
The Ramanujan conjecture
Although there are numerous statements that could have borne the name Ramanujan conjecture, one was highly influential on later work. In particular, the connection of this conjecture with conjectures of
In his paper "On certain arithmetical functions", Ramanujan defined the so-called delta-function, whose coefficients are called τ(n) (the Ramanujan tau function).[117] He proved many congruences for these numbers, such as τ(p) ≡ 1 + p11 mod 691 for primes p. This congruence (and others like it that Ramanujan proved) inspired Jean-Pierre Serre (1954 Fields Medalist) to conjecture that there is a theory of Galois representations that "explains" these congruences and more generally all modular forms. Δ(z) is the first example of a modular form to be studied in this way. Deligne (in his Fields Medal-winning work) proved Serre's conjecture. The proof of Fermat's Last Theorem proceeds by first reinterpreting elliptic curves and modular forms in terms of these Galois representations. Without this theory, there would be no proof of Fermat's Last Theorem.[118]
Ramanujan's notebooks
While still in Madras, Ramanujan recorded the bulk of his results in four notebooks of looseleaf paper. They were mostly written up without any derivations. This is probably the origin of the misapprehension that Ramanujan was unable to prove his results and simply thought up the final result directly. Mathematician Bruce C. Berndt, in his review of these notebooks and Ramanujan's work, says that Ramanujan most certainly was able to prove most of his results, but chose not to record the proofs in his notes.
This may have been for any number of reasons. Since paper was very expensive, Ramanujan did most of his work and perhaps his proofs on slate, after which he transferred the final results to paper. At the time, slates were commonly used by mathematics students in the Madras Presidency. He was also quite likely to have been influenced by the style of G. S. Carr's book, which stated results without proofs. It is also possible that Ramanujan considered his work to be for his personal interest alone and therefore recorded only the results.[119]
The first notebook has 351 pages with 16 somewhat organised chapters and some unorganised material. The second has 256 pages in 21 chapters and 100 unorganised pages, and the third 33 unorganised pages. The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he had found. Hardy himself wrote papers exploring material from Ramanujan's work, as did G. N. Watson, B. M. Wilson, and Bruce Berndt.[119]
In 1976, George Andrews rediscovered a fourth notebook with 87 unorganised pages, the so-called "lost notebook".[101]
Hardy–Ramanujan number 1729
The number 1729 is known as the Hardy–Ramanujan number after a famous visit by Hardy to see Ramanujan at a hospital. In Hardy's words:[120]
I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No", he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."
Immediately before this anecdote, Hardy quoted Littlewood as saying, "Every positive integer was one of [Ramanujan's] personal friends."[121]
The two different ways are:
Generalisations of this idea have created the notion of "taxicab numbers".
Mathematicians' views of Ramanujan
"Of course, we're always hoping. That's one reason I always read letters that come in from obscure places and are written in an illegible scrawl. I always hope it might be from another Ramanujan."
—Freeman Dyson on how another such genius might appear anywhere[122]
In his obituary of Ramanujan, written for Nature in 1920, Hardy observed that Ramanujan's work primarily involved fields less known even among other pure mathematicians, concluding:
His insight into formulae was quite amazing, and altogether beyond anything I have met with in any European mathematician. It is perhaps useless to speculate as to his history had he been introduced to modern ideas and methods at sixteen instead of at twenty-six. It is not extravagant to suppose that he might have become the greatest mathematician of his time. What he actually did is wonderful enough… when the researches which his work has suggested have been completed, it will probably seem a good deal more wonderful than it does to-day.[75]
Hardy further said:[123]
He combined a power of generalisation, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his own peculiar field, without a rival in his day. The limitations of his knowledge were as startling as its profundity. Here was a man who could work out
complex variablewas..."
When asked about the methods Ramanujan employed to arrive at his solutions, Hardy said they were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account."[124] He also said that he had "never met his equal, and can compare him only with Euler or Jacobi".[124] Littlewood reportedly said that helping Ramanujan catch up with European mathematics beyond what was available in India was very difficult, because each new point mentioned to Ramanujan caused him to produce original ideas that prevented Littlewood from continuing the lesson.[125]
K. Srinivasa Rao has said,[126] "As for his place in the world of Mathematics, we quote Bruce C. Berndt: 'Paul Erdős has passed on to us Hardy's personal ratings of mathematicians. Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100. Hardy gave himself a score of 25, J. E. Littlewood 30, David Hilbert 80 and Ramanujan 100.'" During a May 2011 lecture at IIT Madras, Berndt said that over the last 40 years, as nearly all of Ramanujan's conjectures had been proven, there had been greater appreciation of Ramanujan's work and brilliance, and that Ramanujan's work was now pervading many areas of modern mathematics and physics.[115][127]
Posthumous recognition
The year after his death, Nature listed Ramanujan among other distinguished scientists and mathematicians on a "Calendar of Scientific Pioneers" who had achieved eminence.[128] Ramanujan's home state of Tamil Nadu celebrates 22 December (Ramanujan's birthday) as 'State IT Day'. Stamps picturing Ramanujan were issued by the government of India in 1962, 2011, 2012 and 2016.[129]
Since Ramanujan's centennial year, his birthday, 22 December, has been annually celebrated as Ramanujan Day by the
Based on the recommendations of a committee appointed by the University Grants Commission (UGC), Government of India, the Srinivasa Ramanujan Centre, established by SASTRA, has been declared an off-campus centre under the ambit of SASTRA University. House of Ramanujan Mathematics, a museum of Ramanujan's life and work, is also on this campus. SASTRA purchased and renovated the house where Ramanujan lived at Kumabakonam.[130]
In 2011, on the 125th anniversary of his birth, the Indian government declared that 22 December will be celebrated every year as National Mathematics Day.[131] Then Indian Prime Minister Manmohan Singh also declared that 2012 would be celebrated as National Mathematics Year and 22 December as National Mathematics Day of India.[132]
Ramanujan IT City is an information technology (IT) special economic zone (SEZ) in Chennai that was built in 2011. Situated next to the Tidel Park, it includes 25 acres (10 ha) with two zones, with a total area of 5.7 million square feet (530,000 m2), including 4.5 million square feet (420,000 m2) of office space.[133]
Commemorative postal stamps
Commemorative stamps released by India Post (by year):
In popular culture
- The Man Who Loved Numbers is a 1988 PBS NOVA documentary about Ramanujan (S15, E9).[134]
- The Man Who Knew Infinity is a 2015 film based on Kanigel's book of the same name. British actor Dev Patel portrays Ramanujan.[135][136][137]
- Bhama, Kevin McGowan and Michael Lieber star in pivotal roles.[141]
- Nandan Kudhyadi directed the Indian documentary films The Genius of Srinivasa Ramanujan (2013) and Srinivasa Ramanujan: The Mathematician and His Legacy (2016) about the mathematician.[142]
- Ramanujan (The Man Who Reshaped 20th Century Mathematics), an Indian docudrama film directed by Akashdeep released in 2018.[143]
- M. N. Krish's thriller novel The Steradian Trail weaves Ramanujan and his accidental discovery into its plot connecting religion, mathematics, finance and economics.[144][145]
- Partition, a play by Ira Hauptman about Hardy and Ramanujan, was first performed in 2013.[146][147][148][149]
- The play First Class Man by Alter Ego Productions
- Complicite that explores the relationship between Hardy and Ramanujan.[152]
- David Leavitt's novel The Indian Clerk explores the events following Ramanujan's letter to Hardy.[153][154]
- Ramanujan was mentioned in the 1997 film Good Will Hunting, in a scene where professor Gerald Lambeau (Stellan Skarsgård) explains to Sean Maguire (Robin Williams) the genius of Will Hunting (Matt Damon) by comparing him to Ramanujan.[157]
Selected papers
- Ramanujan, S. (1914). "Some definite integrals". Messenger Math. 44: 10–18.
- Ramanujan, S. (1914). "Some definite integrals connected with Gauss's sums". Messenger Math. 44: 75–85.
- Ramanujan, S. (1915). "On certain infinite series". Messenger Math. 45: 11–15.
- Ramanujan, S. (1915). "Highly Composite Numbers". Proceedings of the London Mathematical Society. 14 (1): 347–409. .
- Ramanujan, S. (1915). "On the number of divisors of a number". The Journal of the Indian Mathematical Society. 7 (4): 131–133.
- Ramanujan, S. (1915). "Short Note: On the sum of the square roots of the first n natural numbers". The Journal of the Indian Mathematical Society. 7 (5): 173–175.
- Ramanujan, S. (1916). "Some formulae in the analytical theory of numbers". Messenger Math. 45: 81–84.
- Ramanujan, S. (1916). "A Series for Euler's Constant γ". Messenger Math. 46: 73–80.
- Ramanujan, S. (1917). "On the expression of numbers in the form ax2 + by2 + cz2 + du2". Mathematical Proceedings of the Cambridge Philosophical Society. 19: 11–21.
- Hardy, G. H.; Ramanujan, S. (1917). "Asymptotic Formulae for the Distribution of Integers of Various Types". Proceedings of the London Mathematical Society. 16 (1): 112–132. .
- .
- Hardy, G. H.; Ramanujan, Srinivasa (1918). "On the coefficients in the expansions of certain modular functions". Proc. R. Soc. A. 95 (667): 144–155. .
- Ramanujan, Srinivasa (1919). "Some definite integrals". The Journal of the Indian Mathematical Society. 11 (2): 81–88.
- Ramanujan, S. (1919). "A proof of Bertrand's postulate". The Journal of the Indian Mathematical Society. 11 (5): 181–183.
- Ramanujan, S. (1920). "A class of definite integrals". Quart. J. Pure. Appl. Math. 48: 294–309. .
- Ramanujan, S. (1921). "Congruence properties of partitions". Math. Z. 9 (1–2): 147–153. S2CID 121753215. Posthumously published extract of a longer, unpublished manuscript.
Further works of Ramanujan's mathematics
- ISBN 0-387-25529-X)[158]
- George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part II, (Springer, 2008, ISBN 978-0-387-77765-8)
- George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part III, (Springer, 2012, ISBN 978-1-4614-3809-0)
- George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part IV, (Springer, 2013, ISBN 978-1-4614-4080-2)
- George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part V, (Springer, 2018, ISBN 978-3-319-77832-7)
- M. P. Chaudhary, A simple solution of some integrals given by Srinivasa Ramanujan, (Resonance: J. Sci. Education – publication of Indian Academy of Science, 2008)[159]
- M.P. Chaudhary, Mock theta functions to mock theta conjectures, SCIENTIA, Series A : Math. Sci., (22)(2012) 33–46.
- M.P. Chaudhary, On modular relations for the Roger-Ramanujan type identities, Pacific J. Appl. Math., 7(3)(2016) 177–184.
Selected publications on Ramanujan and his work
- Berndt, Bruce C. (1998). Butzer, P. L.; Oberschelp, W.; Jongen, H. Th. (eds.). Charlemagne and His Heritage: 1200 Years of Civilization and Science in Europe (PDF). Turnhout, Belgium: Brepols Verlag. pp. 119–146. ISBN 978-2-503-50673-9. Archived(PDF) from the original on 9 September 2004.
- Berndt, Bruce C.; Rankin, Robert A. (1995). Ramanujan: Letters and Commentary. Vol. 9. Providence, Rhode Island: ISBN 978-0-8218-0287-8.
- ISBN 978-0-8218-2624-9.
- Berndt, Bruce C. (2006). Number Theory in the Spirit of Ramanujan. Vol. 9. Providence, Rhode Island: ISBN 978-0-8218-4178-5.
- Berndt, Bruce C. (1985). Ramanujan's Notebooks: Part I. New York: Springer. ISBN 978-0-387-96110-1.
- Berndt, Bruce C. (1999). Ramanujan's Notebooks: Part II. New York: Springer. ISBN 978-0-387-96794-3.
- Berndt, Bruce C. (2004). Ramanujan's Notebooks: Part III. New York: Springer. ISBN 978-0-387-97503-0.
- Berndt, Bruce C. (1993). Ramanujan's Notebooks: Part IV. New York: Springer. ISBN 978-0-387-94109-7.
- Berndt, Bruce C. (2005). Ramanujan's Notebooks: Part V. New York: Springer. ISBN 978-0-387-94941-3.
- Hardy, G. H. (March 1937). "The Indian Mathematician Ramanujan". The American Mathematical Monthly. 44 (3): 137–155. JSTOR 2301659.
- Hardy, G. H. (1978). Ramanujan. New York: Chelsea Pub. Co. ISBN 978-0-8284-0136-4.
- Hardy, G. H. (1999). Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. Providence, Rhode Island: American Mathematical Society. ISBN 978-0-8218-2023-0.
- Henderson, Harry (1995). Modern Mathematicians. New York: Facts on File Inc. ISBN 978-0-8160-3235-8.
- ISBN 978-0-684-19259-8.
- ISBN 978-0-7475-9370-6.
- ISBN 978-0-14-303028-7.
- ISBN 978-3319255668.
- Sankaran, T. M. (2005). Srinivasa Ramanujan- Ganitha lokathile Mahaprathibha (Report) (in Malayalam). Kochi, India: Kerala Sasthra Sahithya Parishad.
Selected publications on works of Ramanujan
- Ramanujan, Srinivasa; Hardy, G. H.; Seshu Aiyar, P. V.; ISBN 978-0-8218-2076-6.
- This book was originally published in 1927[160] after Ramanujan's death. It contains the 37 papers published in professional journals by Ramanujan during his lifetime. The third reprint contains additional commentary by Bruce C. Berndt.
- S. Ramanujan (1957). Notebooks (2 Volumes). Bombay: Tata Institute of Fundamental Research.
- These books contain photocopies of the original notebooks as written by Ramanujan.
- S. Ramanujan (1988). The Lost Notebook and Other Unpublished Papers. New Delhi: Narosa. ISBN 978-3-540-18726-4.
- This book contains photo copies of the pages of the "Lost Notebook".
- Problems posed by Ramanujan, Journal of the Indian Mathematical Society.
- S. Ramanujan (2012). Notebooks (2 Volumes). Bombay: Tata Institute of Fundamental Research.
- This was produced from scanned and microfilmed images of the original manuscripts by expert archivists of Roja Muthiah Research Library, Chennai.
See also
- 1729 (number)
- Brown numbers
- List of amateur mathematicians
- List of Indian mathematicians
- Ramanujan graph
- Ramanujan summation
- Ramanujan's constant
- Ramanujan's ternary quadratic form
- Rank of a partition
Footnotes
- ^ FRS (/ˈsriːnɪvɑːsə rɑːˈmɑːnʊdʒən/ SREE-nih-vah-sə rah-MAH-nuuj-ən;[1] born Srinivasa Ramanujan Aiyangar, Tamil: [sriːniʋaːsa ɾaːmaːnud͡ʑan ajːaŋgar])[2][3]
References
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ideas that were critical to the proof of Fermat's last theorem
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External links
Media links
- Biswas, Soutik (16 March 2006). "Film to celebrate mathematics genius". BBC. Retrieved 24 August 2006.
- Feature Film on Mathematics Genius Ramanujan by Dev Benegal and Stephen Fry
- BBC radio programme about Ramanujan – episode 5
- A biographical song about Ramanujan's life
- "Why Did This Mathematician's Equations Make Everyone So Angry?". Youtube.com. Thoughty2. 11 April 2022. Retrieved 29 June 2022.
Biographical links
- Srinivasa Ramanujan at the Mathematics Genealogy Project
- O'Connor, John J.; Robertson, Edmund F., "Srinivasa Ramanujan", MacTutor History of Mathematics Archive, University of St Andrews
- ScienceWorld.
- A short biography of Ramanujan
- "Our Devoted Site for Great Mathematical Genius"
Other links
- Wolfram, Stephen (27 April 2016). "Who Was Ramanujan?".
- A Study Group For Mathematics: Srinivasa Ramanujan Iyengar
- The Ramanujan Journal – An international journal devoted to Ramanujan
- International Math Union Prizes, including a Ramanujan Prize
- Hindu.com: Norwegian and Indian mathematical geniuses, Ramanujan – Essays and Surveys Archived 6 November 2012 at the Wayback Machine, Ramanujan's growing influence, Ramanujan's mentor
- Hindu.com: The sponsor of Ramanujan
- Bruce C. Berndt; Robert A. Rankin (2000). "The Books Studied by Ramanujan in India". MR 1786233.
- "Ramanujan's mock theta function puzzle solved"
- Ramanujan's papers and notebooks
- Sample page from the second notebook
- Ramanujan on Fried Eye
- Clark, Alex. "163 and Ramanujan Constant". Numberphile. Brady Haran. Archived from the original on 4 February 2018. Retrieved 23 June 2018.