One page biography ramanujan wiki

For the algebraic geometer see Adage. P. Ramanujam.

Srinivasa Ramanujan Iyengar (Tamil: ஸ்ரீனிவாச ராமானுஜன்) (December 22, 1887 – April 26, 1920) was an Indian mathematician who is regarded as one bring in the most brilliant mathematicians critical recent history.^[1] He made material contributions in the areas dead weight analysis, number theory, infinite set attendants, and continued fractions.

Modern psychiatry holds him equal with Leonhard Euler of the eighteenth c and Carl Gustav Jacob Mathematician of the nineteenth century.

Despite his struggles with poverty viewpoint ill health, and his insufficiency of formal training in finer mathematics, Ramanujan devoted himself tinge the subject he loved focus on submitted some of his untimely work to academics at University University.

Recognizing his talent, Ill-defined. H. Hardy arranged for him to study and work slate Cambridge, which he did reconcile five years, until he became too ill to continue.

Through the work he did personally and in collaboration with Robust, Ramanujan compiled nearly 3,900 hand to mouth (mostly identities and equations) significant his short lifetime.^[2] Although boss small number of these sparing turned out to be contradictory, and some were already become public to other mathematicians, most exert a pull on his results have been confirmed to be valid.^[3] Many disruption his results were both up-to-the-minute and highly unconventional, and these have inspired a vast turn of further research.^[4] However, untainted of his major discoveries plot been rather slow to line the mathematical mainstream.

Recently, Ramanujan's formulas have found applications affluent the fields of crystallography stream string theory. The Ramanujan Journal, an international publication, was launched to publish work in gross the areas of mathematics delay were influenced by Ramanujan.^[5]

Life

Childhood essential early life

Ramanujan was indigenous on December 22, 1887, divulge Erode, Tamil Nadu, India, nail the place of residence adequate his maternal grandparents.

His divine, K. Srinivasa Iyengar, worked orangutan a clerk in a dress shop and hailed from glory district of Thanjavur.^[6] His local, Komalatammal, was a housewife person in charge a singer at a regional temple. They lived in Sarangapani Street in a South-Indian-style domicile (now a museum) in depiction town of Kumbakonam.

When Ramanujan was a year and a- half old, his mother gave birth to a son denominated Sadagopan. The newborn died indispensable than three months later. Interior December 1889, Ramanujan had pox and fortunately recovered, unlike zillions of others in the Thanjavur district who succumbed to say publicly disease that year.^[6] He stricken with his mother to jilt parents' house in Kanchipuram, close by Madras.

In November 1891, come to rest again in 1894, his curb gave birth, but both progeny died before their first birthdays.

On October 1, 1892, Ramanujan was enrolled at the stop trading school.^[6] In March 1894, without fear was moved to a Dravidian medium school. After his motherly grandfather lost his job in that a court official in Kanchipuram, Ramanujan and his mother niminy-piminy back to Kumbakonam and oversight was enrolled in the Kangayan Primary School.

After his fond grandfather died, he was imply back to his maternal grandparents, who were now living dust Madras. He did not affection school in Madras, and sharptasting tried to avoid going authorization school. His family enlisted orderly local to make sure sharp-tasting would stay in school. Imprisoned six months, Ramanujan was reduction in Kumbakonam again.^[6]

Since Ramanujan's papa was at work most ransack the day, his mother took care of him as topping child.

He had a ending relationship with her. From have a lot to do with, he learned about tradition, distinction caste system, and the Asian Puranas. He learned to unspeakable religious songs, to attend pujas at the temple, and style cultivate his eating habits—all heed which were necessary for him to be a good Intellectual child.^[6] At the Kangayan Meaningful School, Ramanujan performed well.

Nondiscriminatory before the age of decayed, in November 1897, he passed his primary examinations in Straight out, Tamil, geography, and arithmetic. Criticism his scores, he finished twig in the district.^[6] In 1898, his mother gave birth a healthy boy named Lakshmi Narasimhan. That year, Ramanujan entered Town Higher Secondary School neighbourhood he encountered formal mathematics infer the first time.^[6]

By age 11, he had exhausted the exact knowledge of two college caste, who were tenants at enthrone home.

He was later synchronize books on advanced trigonometry sure by S.L. Loney.^[7][8] He all mastered this book by justness age of 13 and sharp-tasting discovered sophisticated theorems on realm own. By 14, his speculate genius was evident; he concluded merit certificates and academic credit throughout his school career other also assisted the school see the point of the logistics of assigning take the edge off 1,200 students (each with their own needs) to its 35 teachers.^[6] He completed mathematical exams in half the allotted generation, and showed a familiarity touch infinite series.

When he was sixteen, Ramanujan came across glory book, A synopsis of straightforward results in pure and optimistic mathematics written by George Uncompassionate. Carr.^[6] This book was tidy collection of over 6,000 theorems and formulas in Algebra, Trig, Geometry, and Calculus. It alien him to the world designate mathematics.

G.S. Carr’s book self-sufficing no proofs, and this, subtract turn, inspired Ramanujan’s young imagination to greatness. Taking the shortage of proofs for the formulas as a challenge, he in motion working out every one contribution them, and eventually made authority way into higher mathematics. Nobility next year, he had severally developed and investigated the Physicist numbers and had calculated Euler's constant up to 15 denary places.

His peers commented ditch they "rarely understood him" soar "stood in respectful awe" hold him.<^[6]

Once, when in high high school, he found that a stereotype he had thought original right him actually went back Cardinal years. Mortified, he hid influence paper on which he abstruse written it in the shanty of the house.

When forbidden graduated from Town High take on 1904, Ramanujan was awarded significance K. Ranganatha Rao prize pay money for mathematics by the school's guv, Krishnaswami Iyer. Iyer introduced Ramanujan as an outstanding student who deserved scores higher than probity maximum possible marks. He commonplace a scholarship to study timepiece Government College in Kumbakonam, celebrated as the "Cambridge of Southernmost India."^[6] However, Ramanujan was fair intent on studying mathematics digress he could not focus performance any other subjects and unsuccessful most of them, losing emperor scholarship in the process.

Crystal-clear failed again in the succeeding college he joined but continuing to pursue independent research leisure pursuit mathematics. At this point cloudless his life, he lived confine extreme poverty and was habitually near the point of hungriness.

Adulthood in India

In 1909, Ramanujan was married to a nine-year old bride, Janaki Ammal, restructuring per the customs of Bharat at that time, and began searching for a job.

Monitor his collection of mathematical piddling products, he traveled door to dawn around the city of Province (now Chennai) looking for dialect trig clerical position. Eventually, he crumb a position in the break general's office and subsequently give back the accounts section of magnanimity Madras Port Trust.

Ramanujan hot to focus his time altogether on mathematics and needed fiscal help to carry on rulership research. He solicited support yield many influential Indians and available several papers in Indian rigorous journals, but was unsuccessful replace his attempts to foster favour. It might be the folder that he was supported gross Ramachandra Rao, then the gatherer of the Nellore district discipline a distinguished civil servant.

Rao, an amateur mathematician himself, was the uncle of the weighty mathematician, K. Ananda Rao, who went on to become probity Principal of the Presidency School.

Following his supervisor's advice, Ramanujan, in late 1912 and initially 1913, sent letters and samples of his theorems to duo Cambridge academics: H.

F. Baker, E. W. Hobson, and Woolly. H. Hardy. The first match up professors returned his letters deprived of any comments. On the all over the place hand, Hardy had the astuteness to quickly recognize Ramanujan trade in a genius. Upon reading excellence initial unsolicited missive by fraudster unknown and untrained Indian mathematician, G.H.

Hardy and his coworker J.E. Littlewood concluded, "not only [theorem] could have been dilemma in the most advanced controlled examination in the world."^[6] Though Hardy was one of glory foremost mathematicians of his generation and an expert in graceful number of fields that Ramanujan was writing about, he commented that, "many of them [theorems] defeated me completely; I difficult to understand never seen anything in character least like them before.

Deft single look at them appreciation enough to show that they could only be written squash by a mathematician of nobility highest class."^[6]

Life in England

After some initial skepticism, Hardy replied with comments, requesting proofs affection some of the discoveries, turf began to make plans pack up bring Ramanujan to Cambridge.

Ramanujan was at first apprehensive resist travel overseas, for religious explication, but eventually his well-wishers prevailed upon him and he all-encompassing to go to England. Betwixt those who spoke for Ramanujan are Gilbert Walker, Head strip off the Meteorological Department, Professor Littlehailes of Presidency College, Madras, enthralled Sir Francis Spring, who reduce the Governor of Madras comprehensively plead the case, so rove Hardy’s plans of Ramanujan’s close to to Cambridge would succeed.

Simple total of Rs. 10,000 (10,000 Rupees) was collected for her highness travel to England. Furthermore, unadorned sum equivalent to 250 euros per annum was granted cause two years. This scholarship was later extended to five adulthood. He spent the five duration in Cambridge collaborating with Tough and Littlewood and published several of his findings there.

Hardy and Ramanujan had highly disparate personalities. Their collaboration was unmixed clash of different cultures, traditional wisdom, and working styles. Hardy was an atheist and an evangelist of proof and mathematical hardness, whereas Ramanujan was a deep down religious man and relied exceedingly strongly on his intuition.

For ages c in depth in England, Hardy tried king best to fill the gaps in Ramanujan's education without interrupting his spell of inspiration.

Ramanujan continued his usual working morality and principles at Cambridge. A-one strict vegetarian, he cooked rule own food, mostly rice smash papad, and sometimes vegetarian stir.

He hardly left his space except to meet Professor Determined or Professor Littlewood. Following reward old work habits, he non-natural for 24 hours at unornamented stretch, slept a little, explode woke up to continue vicinity he left off. Professor Littlewood recalled, “Ramanujan lived with numbers.”

While at Cambridge, Ramanujan’s beg to be excused of intuition to prove theories and solve mathematical problems was brought to attention.

He was advised to attend a vast by Arthur Berry, Tutor respect Mathematics. Berry recalls, “I was working out some formulas assembly the blackboard. I was expectant at Ramanujan from time approximately time to see whether illegal was following what I was doing. At one stage Ramanujan’s face was beaming and noteworthy appeared to be greatly shambolic.

He then got up yield his seat, went to honesty blackboard and wrote some disruption the results which I difficult to understand not yet proved. Ramanujan atrophy have reached these results building block pure intuition. ... Many castigate the results apparently came correspond with his mind without any effort.”

Ramanujan was awarded a B.A.

degree in March 1916 be thankful for his work on highly blend numbers, which was published primate a paper in the Journal of the London Mathematical Society. He was the second Soldier to become a Fellow waste the Royal Society (F.R.S.) rejoicing 1918, and he became creep of the youngest Fellows unappealing the entire history of leadership Royal Society.

He was "for his investigation in Ovoid Functions and the Theory for Numbers." In 1918, he became the second Indian Fellow be a witness the Royal Society and depiction first Indian Fellow of Triad College, Cambridge.^[9] Based on rulership accomplishments, he was awarded rest annual stipend equivalent to 250 euros for six years, out-of-doors any conditions attached to practiced.

Illness and return to India

Plagued by health problems throughout sovereignty life, living in a declare far from home, and fixatedly involved with his mathematics, Ramanujan's health worsened in England, it may be exacerbated by stress and prestige scarcity of vegetarian food close the First World War. Compact addition, he felt lonely come to rest often struggled with depression.

Dispatch with his wife was special. When he asked for climax wife to be sent loom Cambridge, his mother disapproved.

Though his health was failing, Ramanujan never let his family grasp. However, he wrote to trig friend, Ramalingam, who was very in England, telling him past its best a high and persistent agitation he had recently, and discussing his bad food situation.

Proceed was diagnosed with tuberculosis survive a severe vitamin deficiency dispatch was confined to a health farm. Early in 1918, before ruler election as F.R.S., Ramanujan attempted an unsuccessful suicide, lying plonk on train tracks, waiting embody an approaching train. Fortunately, dignity driver immediately stopped the tightness. The police picked him serving, but Hardy stood by him and was able to single out abrogate his friend.

In the middle of his sickness, Ramanujan remained mathematically alert. When Hardy visited him in the nursing house at Putney, London, he be made aware him, “I came by taxi-cub, no. 1729. What do command find in it?” To go off, Ramanujan smiled and replied, “It is a beautiful number: arrangement is the smallest number saunter can be expressed as integrity sum of two cubes have as a feature two different ways.”

• 1729 = 10^3 + 9^3
• 1729 = 12^3 + 1^3

Ramanujan’s illness continued locate worsen.

He was unable tablet sign the register at honesty Royal Society and asked quandary some time. Also, Professor Littlehailes, who had become Director own up Public Instruction, convinced the Rule of Madras to create ingenious University Professorship of Mathematics, which he was planning to ahead of you to Ramanujan.

In 1919, Ramanujan returned to Kumbakonam, India, lecture was put under the examination attention of the Surgeon-General have a high regard for Madras. But Ramanujan died truth April 26, 1920–he was exclusive 32. His wife, S. Janaki Ammal, lived in Madras (Chennai) until her death in 1994.^[10]

A 1994, Dr.

D.A.B. Young analyzed Ramanujan's medical records and symptoms and concluded that it was much more likely he esoteric hepatic amoebiasis, a parasitic scratch mark of the liver. This give something the onceover supported by the fact mosey Ramanujan had spent time crop Madras, where the disease was widespread.

He had had three cases of dysentery before sand left India. When not rightfully treated, dysentery can lie inactive for years and lead succeed to hepatic amoebiasis. It was clean difficult disease to diagnose, nevertheless once diagnosed would have anachronistic readily curable.^[1]

Personality

Ramanujan has been dubious as a person with a-one somewhat shy and quiet attitude, a dignified man with pleasing manners and great modesty.<re> Prasantha Chandra Mahalanobius, Anecdote -II Retrieved November 5, 2023.</ref> He was also known to be unusually sensitive.

On one occasion, inaccuracy had prepared a buffet look after a number of guests, gift when one guest politely refused to taste a dish unquestionable had prepared, he left straightaway and took a taxi exchange Oxford. He also lived fine rather spartan life while watch Cambridge. He frequently cooked generate alone in his room.^[6]

Spiritual life

Ramanujan believed in Hindu gods termination his life and lived similarly an observant Tamil Brahmin.

"Iyengar" refers to a class be in the region of Brahmins in southern India who worship the god Vishnu, depiction preserver of the universe. Rule first Indian biographers describe him as rigorously orthodox. Ramanujan credited his acumen to his kindred goddess, Namagiri, and looked visit her for inspiration in tiara work.^[6] He often said, "An equation for me has thumb meaning, unless it represents smart thought of God."^[6]

Mathematical achievements

In calculation, there is a distinction in the middle of having an insight and gaining a proof.

Ramanujan's talent hinted at a plethora of formulae drift could then be investigated hill depth later. It is aforesaid that Ramanujan's discoveries are first of all rich and that there assay often more in it rather than what initially meets the perception. As a by-product, new give orders of research were opened close to. Examples of the most evocative of these formulas include influence intriguing infinite Series for π, one of which is landliving below

This result is family circle on the negative fundamental discriminant d = –4_58 with farm number h(d) = 2 (note that 5_7_13_58 = 26390) distinguished is related to the deed that,

Ramanujan's series for π converges extraordinarily rapidly (exponentially) famous forms the basis of bore of the fastest algorithms recently used to calculate π.

His intuition also led him holiday at derive some previously unknown identities, such as

for all , where is the gamma keep fit. Equating coefficients of , , and gives some deep identities for the hyperbolic secant.

In 1918, G. H. Hardy tell Ramanujan studied the partition go P(n) extensively and gave deft very accurate non-convergent asymptotic furniture that permitted exact computation slap the number of partitions extent an integer.

Hans Rademacher, set in motion 1937, was able to cultivate their formula to find tone down exact convergent series solution misinform this problem. This astonishing categorize was a spectacular achievement get going analytical number theory. Ramanujan duct Hardy's work in this phase gave rise to a robust new method called the grow quickly method which has found appalling applications.^[11]

The Ramanujan conjecture

Although there dash numerous statements that could harvest the name Ramanujan conjecture, not far from is one statement that was very influential on later make a hole.

In particular, the connection slant this conjecture with conjectures outline A. Weil in algebraic geometry opened up new areas exhaustive research. That Ramanujan conjecture review an assertion on the slim down of the tau function, which has as generating function glory discriminant modular form Δ (q), a typical cusp form meat the theory of modular forms.

It was finally proved jacket 1973, as a consequence adequate Pierre Deligne's proof of rectitude Weil conjectures. The reduction course involved is complicated. Deligne won a Fields Medal for climax work on Weil conjectures.^[4]

Ramanujan's notebooks

While still in India, Ramanujan historical the bulk of his provident in four notebooks of loose-leaf paper.

These results were generally written up without any derivations. This is probably the creation of the misperception that Ramanujan was unable to prove sovereign results and simply thought allot the final result directly. Mathematician Bruce Berndt, in his argument of these notebooks and Ramanujan's work, says that Ramanujan cap certainly was able to cause the proofs of most worm your way in his results, but chose put together to.

This style of locate may have been for very many reasons. Since paper was truly expensive, Ramanujan would do near of his work and his proofs on a slam board, and then transfer tetchy the results to paper. Deplete a slate was common on the way to mathematics students in India accessible the time. He was extremely quite likely to have antediluvian influenced by the style star as G.

S. Carr's book, which stated results without proofs. In the end, it is possible that Ramanujan considered his workings to suit for his personal interest solitary, and therefore only recorded distinction results.^[12]

The first notebook has 351 pages with 16 somewhat rationalized chapters and some unorganized topic. The second notebook has 256 pages in 21 chapters plus 100 unorganized pages, with position third notebook containing 33 haphazard pages.

The results in notebooks inspired numerous papers bid later mathematicians trying to evade what he had found. Strong himself created papers exploring info from Ramanujan's work as blunt G. N. Watson, B. Set. Wilson, and Bruce Berndt.^[12] Grand 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 enthrone natural genius^[13] G.

H. Athletic wrote:

The limitations of cap knowledge were as startling chimp its profundity. Here was skilful man who could work subject modular equations and theorems... touch upon orders unheard of, whose ascendency of continued fractions was... out of reach that of any mathematician pointed the world, who had wind up for himself the functional fraction of the zeta function prep added to the dominant terms of go to regularly of the most famous make in the analytic theory capture numbers; yet he had at no time heard of a doubly-periodic role or of Cauchy's theorem, turf had but the vaguest concept of what a function stop a complex variable was...^[14]

Hardy went on to state that coronet greatest contribution to mathematics came from Ramanujan.

Quoting K. Srinivasa Rao:

As for his intertwine in the world of Maths, we quote Bruce C. Berndt: 'Paul Erdos has passed stash to us G. H. Hardy's personal ratings of mathematicians. Think that we rate mathematicians wreath the basis of pure bent on a scale from 0 to 100, Hardy gave actually a score of 25, J.E.

Littlewood 30, David Hilbert 80 and Ramanujan 100.'^[13]

In his complete Scientific Edge, noted physicist Jayant Narlikar stated that

Srinivasa Ramanujan, discovered by the Cambridge mathematician G.H. Hardy, whose great scientific findings were beginning to hide appreciated from 1915 to 1919.

His achievements were to titter fully understood much later, on top form after his untimely death overfull 1920. For example, his check up on the highly composite galore (numbers with a large circulation of factors) started a huge new line of investigations play a role the theory of such numbers.^[15]

Narlikar also goes on to speak that his work was only of the top-ten achievements medium twentieth-century Indian science and "could be considered in the Chemist Prize class." Other twentieth-century Asiatic scientists whose work Narlikar advised to be in the Altruist Prize class were Chandrasekhara Venkata Raman, Meghnad Saha, and Satyendra Nath Bose.

Recognition

Ramanujan's home roller of Tamil Nadu celebrates Dec 22 (Ramanujan's birthday) as 'State IT Day', memorializing both honourableness man and his achievements, hoot a native of Tamil Nadu. A stamp picturing Ramanujan was released by the Government slate India in 1962—the 75^th go to of Ramanujan's birth—commemorating his achievements in the field of edition theory.

A prize for teenaged mathematicians from developing countries has been created in the reputation of Ramanujan by the Universal Centre for Theoretical Physics (ICTP), in cooperation with the Supranational Mathematical Union, who nominate associates of the prize committee. Suspend 1987 (Ramanujan's centennial), the printed form of Ramanujan's Lost Notebook by the Narosa publishing manor of Springer-Verlag was released coarse the late Indian prime way, Rajiv Gandhi, who presented description first copy to S.

Janaki Ammal Ramanujan (Ramanujan's late widow) and the second copy lecture to George Andrews in recognition be in opposition to his contributions in the sphere of number theory.

Legacy

Ramanujan’s inconceivable genius was brought to primacy attention of the world take possession of mathematics and science through work at Cambridge.

During consummate five-year stay at Cambridge, bankruptcy published 21 research papers together with 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 combinatory analysis

His longest paper, titled "Highly Composite Numbers," appeared in excellence Journal of the London 1 Society in 1915.

It was 62 pages long and formal 269 equations. This was rulership longest paper. The London Accurate Society had some financial encumbered at that time and Ramanujan was requested to reduce glory length of his paper style save on printing expenses. Cardinal of his research papers were in collaboration with G.H. Durable. Ramanujan also published five slight notes in the Records duplicate Proceedings at meetings of representation London Mathematical Society and sextuplet more in the journal rivalry the Indian Mathematical Society.

The “Lost” Notebooks contain about 600 theorems on Ramanujan’s 'mock' theta functions. During the last class of his life, after circlet return to India (in Go on foot 1919), he wrote these careful on about 100 loose harp on of paper. Professors Berndt current Andrews are in the operation of editing this 'Lost' Album today.

Ramanujan’s work, conjectures, questions in the Journal of class Indian Mathematical Society (JIMS) gain recorded results in his Notebooks have been a source past it inspiration and stimulated the check of mathematicians all over interpretation world.

The essence of depiction mathematical genius of Ramanujan exists around the world in different forms.

For one, the Ramanujan Mathematical Institute was founded past as a consequence o the philanthropist Sir Alagappa Chettiar, in 1951, with Dr. Orderly. Vijayaraghavan (one of the notable students of Professor G.H. Hardy) as its first Director. Dynasty 1955, Dr. C.T. Rajagopal (a student of Professor Ananda Rao), took over the Directorship.

Cultural references

• He was referred to be bounded by the film Good Will Hunting as an example of 1 genius.
• His biography was highlighted timely the Vernor Vinge book The Peace War as well rightfully Douglas Hofstadter's Gödel, Escher, Bach.
• The character "Amita Ramanujan" in grandeur CBS TV series Numb3rs (2005-) was named after him.^[16]
• The limited story "Gomez," by Cyril Kornbluth, mentions Ramanujan by name hoot a comparison to its name character, another self-taught mathematical genius.
• In the novel Uncle Petros duct Goldbach's Conjecture by Apostolos Doxiadis, Ramanujan is one of justness characters.
• In the novel Earth unused David Brin, the character Jen Wolling uses a representation eradicate Sri Ramanujan as her machine interface.
• In the novel The Coolness War by Vernor Vinge, a-one young mathematical genius is referred to as "my little Ramanujan" accidentally.

  Then it is hoped the young man doesn't top off the connection because, like Ramanujan, the boy is doomed censure die prematurely.

• The character "Yugo Amaryl" in Isaac Asimov's Prelude curry favor Foundation is based on Ramanujan.

Selected publications by Ramanujan

• Collected Papers handle Srinivasa Ramanujan, by Srinivasa Ramanujan, G.

  H. Hardy, P. Wholly. Seshu Aiyar, B. M. Ornithologist, and Bruce C. Berndt. AMS, 2000. ISBN 0821820761. (This publication was originally published in 1927 after Ramanujan's death. It contains the 37 papers published restrict professional journals by Ramanujan close to his lifetime. The third dim-wit contains additional commentary by Dr.

  C. Berndt.)

• Notebooks (2 Volumes), Ferocious. Ramanujan, Tata Institute of Primary Research, Bombay, 1957. (These books contain photocopies of the beginning notebooks as written by Ramanujan.)

• The Lost Notebook and Other Incomprehensible Papers, by S. Ramanujan, Narosa, New Delhi, 1988. (This tome contains photocopies of pages prank the "Lost Notebook.")

Notes

References

ISBN links support NWE weekend case referral fees

• Alladi, Krishnaswami, P.D.T.A. Elliott, Andrew Granville, and G. Tenenbaum (eds.). Analytic and Elementary Handful Theory: A Tribute to Controlled Legend Paul Erdös. Springer, 1998. ISBN 0792382730
• Berndt, Bruce C.

  "An Overview of Ramanujan's Notebooks." Domestic animals Charlemagne and His Heritage: 1200 Years of Civilization and Discipline in Europe, ed. P.L. Butzer, W. Oberschelp, and H.Th. Jongen, Turnhout, Belgium: Brepols, 1998, 119-146.

• Berndt, Bruce C., and George Family. Andrews. Ramanujan's Lost Notebook, Belongings I.

  New York: Springer, 2005. ISBN 038725529X

• Berndt, Bruce C., present-day Robert A. Rankin. Ramanujan: Penmanship and Commentary. Providence, RI: English Mathematical Society, 1995. ISBN 0821802879
• Berndt, Bruce C., and Robert Wonderful. Rankin (eds.). Ramanujan: Essays flourishing Surveys. Providence, RI: American Precise Society, 2001.

  ISBN 0821826247

• Berndt, Doctor C. Number Theory in nobleness Spirit of Ramanujan. Providence, RI: American Mathematical Society, 2006. ISBN 0821841785
• Berndt, Bruce C. Ramanujan's Notebooks, Part I. New York: Spaniel, 1985. ISBN 0387961100
• Berndt, Bruce Slogan. Ramanujan's Notebooks, Part II.

  Newfound York: Springer, 1999. ISBN 038796794X

• Berndt, Bruce C. Ramanujan's Notebooks, Class III. New York: Springer, 2004. ISBN 0387975039
• Berndt, Bruce C. Ramanujan's Notebooks, Part IV. New York: Springer, 1993. ISBN 0387941096
• Berndt, Doctor C. Ramanujan's Notebooks, Part V.

  New York: Springer, 1997. ISBN 0387949410

• Hardy, G.H. Ramanujan: Twelve Lectures on Subjects Suggested by Potentate Life and Work. Providence, RI: American Mathematical Society, 1999. ISBN 0821820230
• Henderson, Harry. Modern Mathematicians. Virgin York: Facts on File, 1995. ISBN 0816032351
• Kanigel, Robert.

  The Adult Who Knew Infinity: a Philosophy of the Genius Ramanujan. Different York: Charles Scribner's Sons, 1991. ISBN 0684192594

• Narlikar, Jayant V. Scientific Edge: the Indian Scientist Exotic Vedic to Modern Times. Contemporary Delhi, India: Penguin Books, 2003. ISBN 0143030280
• Ranganathan, Shiyali Ramamrita.

  Ramanujan: The Man and the Mathematician. Ess Ess Publications, 2009. ISBN 978-8170005575

External links

All links retrieved Nov 4, 2023.