Srinivasa Ramanujan Iyengar (Tamil: ஸ்ரீனிவாச ராமானுஜன்) (December 22, 1887 - April 26, 1920) was an Indian mathematician who is regarded as one of the most brilliant mathematicians in recent history… 1 He made substantial contributions in the areas of analysis, number theory, infinite series, and continued fractions. Modern analysis holds him equal with Leonhard Euler of the eighteenth century and Carl Gustav Jacob Jacobi of the nineteenth century.
Despite his struggles with poverty and ill health, and his lack of formal training in higher mathematics, Ramanujan devoted himself to the subject he loved and submitted some of his early work to academics at Cambridge University. Recognizing his talent, G. H. Hardy arranged for him to study and work at Cambridge, which he did for five years, until he became too ill to continue.
Through the work he did independently and in collaboration with Hardy, Ramanujan compiled nearly 3,900 results (mostly identities and equations) during his short lifetime.2 Although a small number of these results turned out to be incorrect, and some were already known to other mathematicians, most of his results have been proven to be valid.3 Many of his results were both original and highly unconventional, and these have inspired a vast amount of further research.4 However, some of his major discoveries have been rather slow to enter the mathematical mainstream. Recently, Ramanujan's formulas have found applications in the fields of crystallography and string theory. The Ramanujan Journal, an international publication, was launched to publish work in all the areas of mathematics that were influenced by Ramanujan.5
Childhood and early lifeRamanujan's home on Sarangapani Street, Kumbakonam.
Ramanujan was born on December 22, 1887, in Erode, Tamil Nadu, India, at the place of residence of his maternal grandparents.6 His father, K. Srinivasa Iyengar, worked as a clerk in a sari shop and hailed from the district of Thanjavur.7 His mother, Komalatammal, was a housewife and a singer at a local temple. They lived in Sarangapani Street in a South-Indian-style home (now a museum) in the town of Kumbakonam.
When Ramanujan was a year and a half old, his mother gave birth to a son named Sadagopan. The newborn died less than three months later. In December 1889, Ramanujan had smallpox and fortunately recovered, unlike thousands of others in the Thanjavur district who succumbed to the disease that year.8 He moved with his mother to her parents' house in Kanchipuram, near Madras. In November 1891, and again in 1894, his mother gave birth, but both children died before their first birthdays.
On October 1, 1892, Ramanujan was enrolled at the local school.9 In March 1894, he was moved to a Telugu medium school. After his maternal grandfather lost his job as a court official in Kanchipuram,10 Ramanujan and his mother moved back to Kumbakonam and he was enrolled in the Kangayan Primary School.11 After his paternal grandfather died, he was sent back to his maternal grandparents, who were now living in Madras. He did not like school in Madras, and he tried to avoid going to school. His family enlisted a local to make sure he would stay in school. Within six months, Ramanujan was back in Kumbakonam again.11
Since Ramanujan's father was at work most of the day, his mother took care of him as a child. He had a close relationship with her. From her, he learned about tradition, the caste system, and the Hindu Puranas. He learned to sing religious songs, to attend pujas at the temple, and to cultivate his eating habits-all of which were necessary for him to be a good Brahmin child.12 At the Kangayan Primary School, Ramanujan performed well. Just before the age of ten, in November 1897, he passed his primary examinations in English, Tamil, geography, and arithmetic. With his scores, he finished first in the district.13 In 1898, his mother gave birth to a healthy boy named Lakshmi Narasimhan.8 That year, Ramanujan entered Town Higher Secondary School where he encountered formal mathematics for the first time.14
By age 11, he had exhausted the mathematical knowledge of two college students, who were tenants at his home. He was later lent books on advanced trigonometry written by S.L. Loney.1516 He completely mastered this book by the age of 13 and he discovered sophisticated theorems on his own. By 14, his true genius was evident; he achieved merit certificates and academic awards throughout his school career and also assisted the school in the logistics of assigning its 1,200 students (each with their own needs) to its 35 teachers.17 He completed mathematical exams in half the allotted time, and showed a familiarity with infinite series.
When he was sixteen, Ramanujan came across the book, A synopsis of elementary results in pure and applied mathematics written by George S. Carr.18 This book was a collection of over 6,000 theorems and formulas in Algebra, Trigonometry, Geometry, and Calculus. It introduced him to the world of mathematics. G.S. Carr's book contained no proofs, and this, in turn, inspired Ramanujan's young mind to greatness. Taking the lack of proofs for the formulas as a challenge, he started working out every one of them, and eventually made his way into higher mathematics. The next year, he had independently developed and investigated the Bernoulli numbers and had calculated Euler's constant up to 15 decimal places.19 His peers commented that they "rarely understood him" and "stood in respectful awe" of him.17
Once, when in high school, he found that a formula he had thought original with him actually went back 150 years. Mortified, he hid the paper on which he had written it in the roof of the house.
When he graduated from Town High 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 possible marks.17 He received a scholarship to study at Government College in Kumbakonam,20 known as the "Cambridge of South India."21 However, Ramanujan was so intent on studying mathematics that he could not focus on any other subjects and failed most of them, losing his scholarship in the process. He failed again in the next college he joined but continued to pursue independent research in mathematics. At this point in his life, he lived in extreme poverty and was often near the point of starvation.
Adulthood in India
In 1909, Ramanujan was married to a nine-year old bride, Janaki Ammal, as per the customs of India at that time, and began searching for a job. With his collection of mathematical results, he traveled door to door around the city of Madras (now Chennai) looking for a clerical position. Eventually, he found a position in the accountant general's office and subsequently in the accounts section of the Madras Port Trust. Ramanujan wanted to focus his time completely on mathematics and needed financial help to carry on his research. He solicited support from many influential Indians and published several papers in Indian mathematical journals, but was unsuccessful in his attempts to foster sponsorship. It might be the case that he was supported by Ramachandra Rao, then the collector of the Nellore district and a distinguished civil servant. Rao, an amateur mathematician himself, was the uncle of the well-known mathematician, K. Ananda Rao, who went on to become the Principal of the Presidency College.
Following his supervisor's advice, Ramanujan, in late 1912 and early 1913, sent letters and samples of his theorems to three Cambridge academics: H. F. Baker, E. W. Hobson, and G. H. Hardy. The first two professors returned his letters without any comments. On the other hand, Hardy had the foresight to quickly recognize Ramanujan as a genius. Upon reading the initial unsolicited missive by an unknown and untrained Indian mathematician, G.H. Hardy and his colleague J.E. Littlewood concluded, "not one theorem could have been set in the most advanced mathematical examination in the world."22 Although Hardy was one of the foremost mathematicians of his day and an expert in a number of fields that Ramanujan was writing about, he commented that, "many of them theorems defeated me completely; I had never seen anything in the least like them before. A single look at them is enough to show that they could only be written down by a mathematician of the highest class."22
Life in England
After some initial skepticism, Hardy replied with comments, requesting proofs for some of the discoveries, and began to make plans to bring Ramanujan to Cambridge. Ramanujan was at first apprehensive to travel overseas, for religious reasons, but eventually his well-wishers prevailed upon him and he agreed to go to England. Among those who spoke for Ramanujan are Gilbert Walker, Head of the Meteorological Department, Professor Littlehailes of Presidency College, Madras, and Sir Francis Spring, who met the Governor of Madras to plead the case, so that Hardy's plans of Ramanujan's coming to Cambridge would succeed. A total of Rs. 10,000 (10,000 Rupees) was collected for his travel to England. Furthermore, a sum equivalent to 250 euros per annum was granted for two years. This scholarship was later extended to five years. He spent the five years in Cambridge collaborating with Hardy and Littlewood and published some of his findings there.
Hardy and Ramanujan had highly contrasting personalities. Their collaboration was a clash of different cultures, beliefs, and working styles. Hardy was an atheist and an apostle of proof and mathematical rigor, whereas Ramanujan was a deeply religious man and relied very strongly on his intuition. While in England, Hardy tried his best to fill the gaps in Ramanujan's education without interrupting his spell of inspiration.
Ramanujan continued his usual working habits and principles at Cambridge. A strict vegetarian, he cooked his own food, mostly rice with papad, and sometimes vegetarian soup. He hardly left his room except to meet Professor Hardy or Professor Littlewood. Following his old work habits, he worked for 24 hours at a stretch, slept a little, and woke up to continue where he left off. Professor Littlewood recalled, “Ramanujan lived with numbers.”
While at Cambridge, Ramanujan's use of intuition to prove theories and solve mathematical problems was brought to attention. He was advised to attend a class by Arthur Berry, Tutor in Mathematics. Berry recalls, “I was working out some formulas on the blackboard. I was looking at Ramanujan from time to time to see whether he was following what I was doing. At one stage Ramanujan's face was beaming and he appeared to be greatly excited. He then got up from his seat, went to the blackboard and wrote some of the results which I had not yet proved. Ramanujan must have reached these results by pure intuition… Many of the results apparently came to his mind without any effort.”
Ramanujan was awarded a B.A. degree in March 1916 for his work on highly composite numbers, which was published as a paper in the Journal of the London Mathematical Society. He was the second Indian to become a Fellow of the Royal Society (F.R.S.) in 1918, and he became one of the youngest Fellows in the entire history of the Royal Society. He was elected "for his investigation in Elliptic Functions and the Theory of Numbers." On October 13, 1918, he became the first Indian to be elected a Fellow of Trinity College, Cambridge.23 Based on his accomplishments, he was awarded an annual stipend equivalent to 250 euros for six years, without any conditions attached to it.
Illness and return to India
Plagued by health problems throughout his life, living in a country far from home, and obsessively involved with his mathematics, Ramanujan's health worsened in England, perhaps exacerbated by stress and the scarcity of vegetarian food during the First World War. In addition, he felt lonely and often struggled with depression. Correspondence with his wife was irregular. When he asked for his wife to be sent to Cambridge, his mother disapproved.
Though his health was failing, Ramanujan never let his family know. However, he wrote to a friend, Ramalingam, who was also in England, telling him of a high and persistent fever he had recently, and discussing his bad food situation. He was diagnosed with tuberculosis and a severe vitamin deficiency and was confined to a sanatorium. Early in 1918, before his election as F.R.S., Ramanujan attempted an unsuccessful suicide, lying down on train tracks, waiting for an approaching train. Fortunately, the driver immediately stopped the train. The police picked him up, but Hardy stood by him and was able to save his friend.
In the midst of his sickness, Ramanujan remained mathematically alert. When Hardy visited him in the nursing home at Putney, London, he told him, “I came by taxi, no. 1729. What do you find in it?” To that, Ramanujan smiled and replied, “It is a beautiful number: it is the smallest number that can be expressed as the sum of two cubes in two different ways.”
- 1729 = 10^3 + 9^3
- 1729 = 12^3 + 1^3
Ramanujan's illness continued to worsen. He was unable to sign the register at the Royal Society and asked for some time. Also, Professor Littlehailes, who had become Director of Public Instruction, convinced the University of Madras to create a University Professorship of Mathematics, which he was planning to offer to Ramanujan.
In 1919, Ramanujan returned to Kumbakonam, India, and was put under the medical attention of the Surgeon-General of Madras. But Ramanujan died on April 26, 1920-he was only 32. His wife, S. Janaki Ammal, lived in Madras (Chennai) until her death in 1994.24
A 1994, Dr. D.A.B. Young analyzed Ramanujan's medical records and symptoms and concluded that it was much more likely he had hepatic amoebiasis, a parasitic infection of the liver. This is supported by the fact that Ramanujan had spent time in Madras, where the disease was widespread. He had had two cases of dysentery before he left India. When not properly treated, dysentery can lie dormant for years and lead to hepatic amoebiasis.1 It was a difficult disease to diagnose, but once diagnosed would have been readily curable.1
Ramanujan has been described as a person with a somewhat shy and quiet disposition, a dignified man with pleasant manners and great modesty.25 He was also known to be extremely sensitive. On one occasion, he had prepared a buffet for a number of guests, and when one guest politely refused to taste a dish he had prepared, he left immediately and took a taxi to Oxford. He also lived a rather spartan life while at Cambridge. He frequently cooked vegetables alone in his room.22
Ramanujan believed in Hindu gods all his life and lived as an observant Tamil Brahmin. "Iyengar" refers to a class of Brahmins in southern India who worship the god Vishnu, the preserver of the universe. His first Indian biographers describe him as rigorously orthodox. Ramanujan credited his acumen to his family goddess, Namagiri, and looked to her for inspiration in his work.26 He often said, "An equation for me has no meaning, unless it represents a thought of God."27
In mathematics, there is a distinction between having an insight and having a proof. Ramanujan's talent suggested a plethora of formulae that could then be investigated in depth later. It is said that Ramanujan's discoveries are unusually rich and that there is often more in it than what initially meets the eye. As a by-product, new directions of research were opened up. Examples of the most interesting of these formulas include the intriguing infinite Series for π, one of which is given below
This result is based on the negative fundamental discriminant d = -4_58 with class number h(d) = 2 (note that 5_7_13_58 = 26390) and is related to the fact that,
Ramanujan's series for π converges extraordinarily rapidly (exponentially) and forms the basis of some of the fastest algorithms currently used to calculate π.
His intuition also led him to derive some previously unknown identities, such as
for all , where is the gamma function. Equating coefficients of , , and gives some deep identities for the hyperbolic secant.
In 1918, G. H. Hardy and Ramanujan studied the partition function P (n) extensively and gave a very accurate non-convergent asymptotic series that permitted exact computation of the number of partitions of an integer. Hans Rademacher, in 1937, was able to refine their formula to find an exact convergent series solution to this problem. This astonishing formula was a spectacular achievement in analytical number theory. Ramanujan and Hardy's work in this area gave rise to a powerful new method called the circle method which has found tremendous applications.28
The Ramanujan conjecture
Although there are numerous statements that could bear the name Ramanujan conjecture, there is one statement that was very influential on later work. In particular, the connection of this conjecture with conjectures of A. Weil in algebraic geometry opened up new areas of research. That Ramanujan conjecture is an assertion on the size of the tau function, which has as generating function the discriminant modular form Δ (q), a typical cusp form in the theory of modular forms. It was finally proved in 1973, as a consequence of Pierre Deligne's proof of the Weil conjectures. The reduction step involved is complicated. Deligne won a Fields Medal for his work on Weil conjectures.29
While still in India, Ramanujan recorded the bulk of his results in four notebooks of loose-leaf paper. These results were mostly written up without any derivations. This is probably the origin of the misperception that Ramanujan was unable to prove his results and simply thought up the final result directly. Mathematician Bruce Berndt, in his review of these notebooks and Ramanujan's work, says that Ramanujan most certainly was able to make the proofs of most of his results, but chose not to.
This style of working may have been for several reasons. Since paper was very expensive, Ramanujan would do most of his work and perhaps his proofs on a slate board, and then transfer just the results to paper. Using a slate was common for mathematics students in India at the time. He was also quite likely to have been influenced by the style of G. S. Carr's book, which stated results without proofs. Finally, it is possible that Ramanujan considered his workings to be for his personal interest alone, and therefore only recorded the results.30
The first notebook has 351 pages with 16 somewhat organized chapters and some unorganized material. The second notebook has 256 pages in 21 chapters and 100 unorganized pages, with the third notebook containing 33 unorganized pages. The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he had found. Hardy himself created papers exploring material from Ramanujan's work as did G. N. Watson, B. M. Wilson, and Bruce Berndt.30 A fourth notebook, the so-called "lost notebook", was rediscovered in 1976 by George Andrews.1
Other mathematicians' views of Ramanujan
Ramanujan is generally hailed as an all-time great mathematician, in the league of Leonhard Euler, Johann Gauss, and Carl Gustav Jacob Jacobi, for his natural genius31 G. H. Hardy quotes: "The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems… to orders unheard of, whose mastery of continued fractions was… beyond that of any mathematician in the world, who had found for himself the functional equation of the zeta function and the dominant terms of many of the most famous problems in the analytic theory of numbers; yet he had never heard of a doubly-periodic function or of Cauchy's theorem, and had but the vaguest idea of what a function of a complex variable was… "32 Hardy went on to state that his greatest contribution to mathematics came from Ramanujan.
Quoting K. Srinivasa Rao,33 "As for his place in the world of Mathematics, we quote Bruce C. Berndt: 'Paul Erd_s has passed on to us G. H. 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.'"
In his book Scientific Edge, noted physicist Jayant Narlikar stated that "Srinivasa Ramanujan, discovered by the Cambridge mathematician G.H. Hardy, whose great mathematical findings were beginning to be appreciated from 1915 to 1919. His achievements were to be fully understood much later, well after his untimely death in 1920. For example, his work on the highly composite numbers (numbers with a large number of factors) started a whole new line of investigations in the theory of such numbers." Narlikar also goes on to say that his work was one of the top-ten achievements of twentieth-century Indian science and "could be considered in the Nobel Prize class." Other twentieth-century Indian scientists whose work Narlikar considered to be in the Nobel Prize class were Chandrasekhara Venkata Raman, Meghnad Saha, and Satyendra Nath Bose.
Ramanujan's home state of Tamil Nadu celebrates December 22 (Ramanujan's birthday) as 'State IT Day', memorializing both the man and his achievements, as a native of Tamil Nadu. A stamp picturing Ramanujan was released by the Government of India in 1962-the 75th anniversary of Ramanujan's birth-commemorating his achievements in the field of number theory.
A prize for young mathematicians from developing countries has been created in the name of Ramanujan by the International Centre for Theoretical Physics (ICTP), in cooperation with the International Mathematical Union, who nominate members of the prize committee. In 1987 (Ramanujan's centennial), the printed form of Ramanujan's Lost Notebook by the Narosa publishing house of Springer-Verlag was released by the late Indian prime minister, Rajiv Gandhi, who presented the first copy to S. Janaki Ammal Ramanujan (Ramanujan's late widow) and the second copy to George Andrews in recognition of his contributions in the field of number theory.
Ramanujan's incredible genius was brought to the attention of the world of mathematics and science through his work at Cambridge. During his five-year stay at Cambridge, he published 21 research papers containing theorems on the following topics:
- Definite integral
- Modular equations and functions
- Riemann's zeta function
- Infinite series
- Summation of series
- Analytic number theory
- Asymptotic formulae
- Partitions and combinatorial analysis
His longest paper, titled "Highly Composite Numbers," appeared in the Journal of the London Mathematical Society in 1915. It was 62 pages long and contained 269 equations. This was his longest paper. The London Mathematical Society had some financial difficulties at that time and Ramanujan was requested to reduce the length of his paper to save on printing expenses. Seven of his research papers were in collaboration with G.H. Hardy. Ramanujan also published five short notes in the Records of Proceedings at meetings of the London Mathematical Society and six more in the journal of the Indian Mathematical Society.
The “Lost” Notebooks contain about 600 theorems on Ramanujan's 'mock' theta functions. During the last year of his life, after his return to India (in March 1919), he wrote these results on about 100 loose sheets of paper. Professors Berndt and Andrews are in the process of editing this 'Lost' Notebook today.
Ramanujan's work, conjectures, questions in the Journal of the Indian Mathematical Society (JIMS) and recorded results in his Notebooks have been a source of inspiration and stimulated the research of mathematicians all over the world.
The essence of the mathematical genius of Ramanujan exists around the world in various forms. For one, the Ramanujan Mathematical Institute was founded by the philanthropist Sir Alagappa Chettiar, in 1951, with Dr. T. Vijayaraghavan (one of the talented students of Professor G.H. Hardy) as its first Director. In 1955, Dr. C.T. Rajagopal (a student of Professor Ananda Rao), took over the Directorship.
- An international feature film on Ramanujan's life will begin shooting in 2007 in Tamil Nadu state and Cambridge. It is being produced by an Indo-British collaboration; it will be co-directed by Stephen Fry and Dev Benegal. 34 A play First Class Man by Alter Ego Productions 35 was based on David Freeman's "First Class Man." The play is centered on Ramanujan and his relationship with G. H. Hardy.
- Another film based on the book The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel is being made by Edward Pressman and Matthew Brown.36
- He was referred to in the film Good Will Hunting as an example of mathematical genius.
- His biography was highlighted in the Vernor Vinge book The Peace War as well as Douglas Hofstadter's Gödel, Escher, Bach.
- The character "Amita Ramanujan" in the CBS TV series Numb3rs (2005-) was named after him.37
- The short story "Gomez," by Cyril Kornbluth, mentions Ramanujan by name as a comparison to its title character, another self-taught mathematical genius.
- In the novel Uncle Petros and Goldbach's Conjecture by Apostolos Doxiadis, Ramanujan is one of the characters.
- In the novel Earth by David Brin, the character Jen Wolling uses a representation of Sri Ramanujan as her computer interface.
- In the novel The Peace War by Vernor Vinge, a young mathematical genius is referred to as "my little Ramanujan" accidentally. Then it is hoped the young man doesn't get the connection because, like Ramanujan, the boy is doomed to die prematurely.
- The character "Yugo Amaryl" in Isaac Asimov's Prelude to Foundation is based on Ramanujan.
- The theatre company Complicite38 has created a production based around the life of Ramanjuan called A Disappearing Number39-conceived and directed by Simon McBurney
- Infinite series
- Number theory
Selected publications by Ramanujan
- Collected Papers of Srinivasa Ramanujan, by Srinivasa Ramanujan, G. H. Hardy, P. V. Seshu Aiyar, B. M. Wilson, and Bruce C. Berndt. AMS, 2000. ISBN 0821820761. (This book was originally published in 1927 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.)
- Notebooks (2 Volumes), S. Ramanujan, Tata Institute of Fundamental Research, Bombay, 1957. (These books contain photocopies of the original notebooks as written by Ramanujan.)
- The Lost Notebook and Other Unpublished Papers, by S. Ramanujan, Narosa, New Delhi, 1988. (This book contains photocopies of pages in the "Lost Notebook.")
- ↑ 1.0 1.1 1.2 1.3 Peterson, Doug. Raiders of the Lost Notebook. UIUC College of Liberal Arts and Sciences. Retrieved February 10, 2009.
- ↑ Berndt, Bruce C. (2005). Ramanujan's Notebooks Part V. SpringerLink, 4. ISBN 0387949410.
- ↑ (August 1999)Rediscovering Ramanujan. Frontline 16 (17): 650., Retrieved February 10, 2009
- ↑ Ono, Ken (June-July 2006). Honoring a Gift from Kumbakonam. Notices of the American Mathematical Society 53 (6): 650., Retrieved February 11, 2009.
- ↑ Alladi, Krishnaswami (1998). Analytic and Elementary Number Theory: A Tribute to Mathematical Legend Paul Erdös. Norwell, Massachusetts: Kluwer Academic Publishers, 6. ISBN 0792382730.
- ↑ Kanigel, Robert (1991). The Man Who Knew Infinity: A Life of the Genius Ramanujan. New York: Charles Scribner's Sons, 11. ISBN 0684192594.
- ↑ Kanigel 1991, 17-18.
- ↑ 8.0 8.1 Kanigel 1991, 12.
- ↑ Kanigel 1991, 13.
- ↑ Kanigel 1991, 19.
- ↑ 11.0 11.1 Kanigel 1991, 14.
- ↑ Kanigel 1991, 20.
- ↑ Kanigel 1991, 25.
- ↑ Kanigel 1991, 25.
- ↑ Hardy, G. H. (1999). Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. Providence, Rhode Island: American Mathematical Society, 2. ISBN 0821820230.
- ↑ Berndt, Bruce C. and Robert A. Rankin (2001). Ramanujan: Essays and Surveys. Providence, Rhode Island: American Mathematical Society, 9. ISBN 0821826247.
- ↑ 17.0 17.1 17.2 Kanigel 1991, 27.
- ↑ Kanigel 1991, 39.
- ↑ Kanigel 1991, 90.
- ↑ Kanigel 1991, 28.
- ↑ Kanigel 1991, 45.
- ↑ 22.0 22.1 22.2 The man who knew infinity.
- ↑ Srinivasa Ramanujan: A Remarkable Mathematical Genius., Retrieved February 11, 2009
- ↑ Ramanujan's wife: Janakiammal (Janaki)., Retrieved February 11, 2009
- ↑ Ramanujan's Pers