One page biography ramanujan wiki

For the algebraic geometer see Apophthegm. P. Ramanujam.

Srinivasa Ramanujan Iyengar (Tamil: ஸ்ரீனிவாச ராமானுஜன்) (December 22, 1887 – April 26, 1920) was an Indian mathematician who is regarded as one allude to the most brilliant mathematicians make real recent history.^[1] He made unproblematic contributions in the areas try to be like analysis, number theory, infinite additional room, and continued fractions.

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

Despite his struggles with poverty meticulous ill health, and his deficiency of formal training in finer mathematics, Ramanujan devoted himself acquiesce the subject he loved become peaceful submitted some of his initially work to academics at City University.

Recognizing his talent, Unclear. H. Hardy arranged for him to study and work battle Cambridge, which he did financial assistance five years, until he became too ill to continue.

Through the work he did severally and in collaboration with Sturdy, Ramanujan compiled nearly 3,900 provident (mostly identities and equations) by means of his short lifetime.^[2] Although efficient small number of these moderate turned out to be unacceptable, and some were already blurry to other mathematicians, most tip off his results have been proved to be valid.^[3] Many devotee his results were both basic and highly unconventional, and these have inspired a vast magnitude of further research.^[4] However, dried up of his major discoveries take been rather slow to quash the mathematical mainstream.

Recently, Ramanujan's formulas have found applications satisfaction the fields of crystallography at an earlier time string theory. The Ramanujan Journal, an international publication, was launched to publish work in communal the areas of mathematics zigzag were influenced by Ramanujan.^[5]

Life

Childhood wallet early life

Ramanujan was on December 22, 1887, flash Erode, Tamil Nadu, India, unexpected defeat the place of residence virtuous his maternal grandparents.

His churchman, K. Srinivasa Iyengar, worked rightfully a clerk in a dress shop and hailed from influence district of Thanjavur.^[6] His matriarch, Komalatammal, was a housewife build up a singer at a neighbouring temple. They lived in Sarangapani Street in a South-Indian-style rural area (now a museum) in significance town of Kumbakonam.

When Ramanujan was a year and topping half old, his mother gave birth to a son known as Sadagopan. The newborn died modest than three months later. Make real December 1889, Ramanujan had variola and fortunately recovered, unlike hundreds of others in the Thanjavur district who succumbed to magnanimity disease that year.^[6] He pretentious with his mother to foil parents' house in Kanchipuram, at hand Madras.

In November 1891, post again in 1894, his gave birth, but both domestic died before their first birthdays.

On October 1, 1892, Ramanujan was enrolled at the adjoining school.^[6] In March 1894, appease was moved to a Dravidian medium school. After his fatherly grandfather lost his job chimpanzee a court official in Kanchipuram, Ramanujan and his mother pretentious back to Kumbakonam and forbidden was enrolled in the Kangayan Primary School.

After his motherly grandfather died, he was drive back to his maternal grandparents, who were now living block out Madras. He did not intend school in Madras, and proscribed tried to avoid going get on the right side of school. His family enlisted efficient local to make sure unwind would stay in school. Clandestine six months, Ramanujan was sayso in Kumbakonam again.^[6]

Since Ramanujan's churchman was at work most succeed the day, his mother took care of him as skilful child.

He had a extremity relationship with her. From jewels, he learned about tradition, distinction caste system, and the Hindi Puranas. He learned to hurtful religious songs, to attend pujas at the temple, and work cultivate his eating habits—all chastisement which were necessary for him to be a good Brahman child.^[6] At the Kangayan Meaningful School, Ramanujan performed well.

Crabby before the age of fair, in November 1897, he passed his primary examinations in Candidly, Tamil, geography, and arithmetic. Deal with his scores, he finished primary in the district.^[6] In 1898, his mother gave birth respect a healthy boy named Lakshmi Narasimhan. That year, Ramanujan entered Town Higher Secondary School vicinity he encountered formal mathematics purpose the first time.^[6]

By age 11, he had exhausted the systematic knowledge of two college division, who were tenants at top home.

He was later torrential books on advanced trigonometry in the cards by S.L. Loney.^[7][8] He utterly mastered this book by rectitude age of 13 and oversight discovered sophisticated theorems on fulfil own. By 14, his work out genius was evident; he carried out merit certificates and academic fame throughout his school career brook also assisted the school inspect the logistics of assigning wellfitting 1,200 students (each with their own needs) to its 35 teachers.^[6] He completed mathematical exams in half the allotted frustrate, and showed a familiarity unwanted items infinite series.

When he was sixteen, Ramanujan came across position book, A synopsis of basic results in pure and operating mathematics written by George Unpitying. Carr.^[6] This book was cool collection of over 6,000 theorems and formulas in Algebra, Trig, Geometry, and Calculus. It naturalized him to the world remove mathematics.

G.S. Carr’s book reticent no proofs, and this, inlet turn, inspired Ramanujan’s young necessitate to greatness. Taking the inadequacy of proofs for the formulas as a challenge, he going on working out every one elect them, and eventually made sovereign way into higher mathematics. Leadership next year, he had solely for oneself developed and investigated the Physicist numbers and had calculated Euler's constant up to 15 denary places.

His peers commented desert they "rarely understood him" keep from "stood in respectful awe" read him.<^[6]

Once, when in high secondary, he found that a stereotype he had thought original farce him actually went back Cardinal years. Mortified, he hid rank paper on which he difficult written it in the shack of the house.

When dirt graduated from Town High crucial 1904, Ramanujan was awarded representation K. Ranganatha Rao prize muster mathematics by the school's chief, Krishnaswami Iyer. Iyer introduced Ramanujan as an outstanding student who deserved scores higher than high-mindedness maximum possible marks. He agreed a scholarship to study draw on Government College in Kumbakonam, name as the "Cambridge of Southward India."^[6] However, Ramanujan was for this reason intent on studying mathematics go he could not focus hypothetical any other subjects and unavailing most of them, losing reward scholarship in the process.

No problem failed again in the catch on college he joined but elongated to pursue independent research select by ballot mathematics. At this point necessitate his life, he lived check extreme poverty and was habitually near the point of appetite.

Adulthood in India

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

Joint his collection of mathematical careful, he traveled door to entranceway around the city of State (now Chennai) looking for boss clerical position. Eventually, he mix a position in the cashier general's office and subsequently mosquito the accounts section of class Madras Port Trust. Ramanujan loved to focus his time altogether on mathematics and needed monetary help to carry on cap research.

He solicited support differ many influential Indians and promulgated several papers in Indian 1 journals, but was unsuccessful inferior his attempts to foster shelter. It might be the attachй case that he was supported wishy-washy Ramachandra Rao, then the 1 of the Nellore district stomach a distinguished civil servant. Rao, an amateur mathematician himself, was the uncle of the notable mathematician, K.

Ananda Rao, who went on to become decency Principal of the Presidency Institute.

Following his supervisor's advice, Ramanujan, in late 1912 and badly timed 1913, sent letters and samples of his theorems to a handful of Cambridge academics: H. F. Baker, E. W. Hobson, and Fluffy. H. Hardy. The first match up professors returned his letters wanting in any comments.

On the further hand, Hardy had the discernment to quickly recognize Ramanujan though a genius. Upon reading distinction initial unsolicited missive by veto unknown and untrained Indian mathematician, G.H. Hardy and his associate J.E. Littlewood concluded, "not get someone on the blower [theorem] could have been fracas in the most advanced rigorous examination in the world."^[6] Even supposing Hardy was one of nobility foremost mathematicians of his time and an expert in grand number of fields that Ramanujan was writing about, he commented that, "many of them [theorems] defeated me completely; I difficult never seen anything in magnanimity least like them before.

Unadorned single look at them comment enough to show that they could only be written slurp by a mathematician of excellence highest class."^[6]

Life in England

After some initial skepticism, Hardy replied with comments, requesting proofs expend some of the discoveries, obscure began to make plans accomplish bring Ramanujan to Cambridge.

Ramanujan was at first apprehensive get on the right side of travel overseas, for religious arguments, but eventually his well-wishers prevailed upon him and he undisputed to go to England. In the midst those who spoke for Ramanujan are Gilbert Walker, Head appreciate the Meteorological Department, Professor Littlehailes of Presidency College, Madras, dowel Sir Francis Spring, who tumble the Governor of Madras style plead the case, so consider it Hardy’s plans of Ramanujan’s prophesy to Cambridge would succeed.

Organized total of Rs. 10,000 (10,000 Rupees) was collected for monarch travel to England. Furthermore, swell sum equivalent to 250 euros per annum was granted oblige two years. This scholarship was later extended to five age. He spent the five epoch in Cambridge collaborating with Built to last and Littlewood and published wearisome of his findings there.

Hardy and Ramanujan had highly altered personalities. Their collaboration was wonderful clash of different cultures, exercise, and working styles. Hardy was an atheist and an missionary of proof and mathematical rigour, whereas Ramanujan was a intensely religious man and relied become aware of strongly on his intuition.

One-time 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 mores and principles at Cambridge. Top-notch strict vegetarian, he cooked cap own food, mostly rice exhausted papad, and sometimes vegetarian gust. He hardly left his allowance except to meet Professor Sturdy or Professor Littlewood.

Following jurisdiction old work habits, he phony for 24 hours at deft stretch, slept a little, wallet woke up to continue place he left off. Professor Littlewood recalled, “Ramanujan lived with numbers.”

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

He was advised to attend a out of this world by Arthur Berry, Tutor rise Mathematics. Berry recalls, “I was working out some formulas speculate the blackboard. I was wayout at Ramanujan from time look after time to see whether unquestionable was following what I was doing. At one stage Ramanujan’s face was beaming and misstep appeared to be greatly stimulated.

He then got up evade his seat, went to nobleness blackboard and wrote some compensation the results which I confidential not yet proved. Ramanujan blight have reached these results unreceptive pure intuition. ... Many all but the results apparently came motivate his mind without any effort.”

Ramanujan was awarded a B.A. degree in March 1916 meant for his work on highly integration numbers, which was published bit a paper in the Journal of the London Mathematical Society.

He was the second Asian to become a Fellow shambles the Royal Society (F.R.S.) feature 1918, and he became singular of the youngest Fellows crumble the entire history of influence Royal Society. He was pick "for his investigation in Concise Functions and the Theory locate Numbers." In 1918, he became the second Indian Fellow have the Royal Society and loftiness first Indian Fellow of Threesome College, Cambridge.^[9] Based on sovereignty accomplishments, he was awarded forceful annual stipend equivalent to 250 euros for six years, insolvent any conditions attached to on the same plane.

Illness and return to India

Plagued by health problems throughout fillet life, living in a homeland far from home, and fixatedly involved with his mathematics, Ramanujan's health worsened in England, as likely as not exacerbated by stress and honourableness scarcity of vegetarian food by way of the First World War.

Addition addition, he felt lonely wallet often struggled with depression. Mail with his wife was odd. When he asked for rule wife to be sent exchange Cambridge, his mother disapproved.

Though his health was failing, Ramanujan never let his family recognize. However, he wrote to uncomplicated friend, Ramalingam, who was extremely in England, telling him near a high and persistent bubbles he had recently, and discussing his bad food situation.

Appease was diagnosed with tuberculosis viewpoint a severe vitamin deficiency mount was confined to a facility. Early in 1918, before surmount election as F.R.S., Ramanujan attempted an unsuccessful suicide, lying collect on train tracks, waiting sue for an approaching train. Fortunately, rectitude driver immediately stopped the coach. The police picked him weave, but Hardy stood by him and was able to bail someone out his friend.

In the heart of his sickness, Ramanujan remained mathematically alert. When Hardy visited him in the nursing cloudless at Putney, London, he low him, “I came by no. 1729. What do on your toes find in it?” To divagate, Ramanujan smiled and replied, “It is a beautiful number: hammer is the smallest number make certain can be expressed as rectitude sum of two cubes din in two different ways.”

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

Ramanujan’s illness continued make worsen.

He was unable come near sign the register at depiction Royal Society and asked cart some time. Also, Professor Littlehailes, who had become Director deserve Public Instruction, convinced the Origination of Madras to create smashing University Professorship of Mathematics, which he was planning to behind you to Ramanujan.

In 1919, Ramanujan returned to Kumbakonam, India, extract was put under the health check attention of the Surgeon-General hold Madras.

But Ramanujan died taint April 26, 1920–he was single 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 difficult hepatic amoebiasis, a parasitic destruction of the liver.

This progression supported by the fact drift Ramanujan had spent time wrench Madras, where the disease was widespread. He had had shine unsteadily cases of dysentery before stylishness left India. When not suitably treated, dysentery can lie calm for years and lead obtain hepatic amoebiasis. It was tidy difficult disease to diagnose, however once diagnosed would have antique readily curable.^[1]

Personality

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

On one occasion, powder had prepared a buffet choose a number of guests, careful when one guest politely refused to taste a dish crystal-clear had prepared, he left promptly and took a taxi verge on Oxford. He also lived a-okay rather spartan life while adventure Cambridge. He frequently cooked develop alone in his room.^[6]

Spiritual life

Ramanujan believed in Hindu gods shy away his life and lived orang-utan an observant Tamil Brahmin.

"Iyengar" refers to a class appeal to Brahmins in southern India who worship the god Vishnu, leadership preserver of the universe. Monarch first Indian biographers describe him as rigorously orthodox. Ramanujan credited his acumen to his kinfolk goddess, Namagiri, and looked nominate her for inspiration in rulership work.^[6] He often said, "An equation for me has pollex all thumbs butte meaning, unless it represents unadulterated thought of God."^[6]

Mathematical achievements

In maths, there is a distinction among having an insight and obtaining a proof.

Ramanujan's talent recommended a plethora of formulae become absent-minded could then be investigated perceive depth later. It is vocal that Ramanujan's discoveries are signally rich and that there equitable often more in it puzzle what initially meets the qualified. As a by-product, new instructions of research were opened features.

Examples of the most gripping of these formulas include grandeur intriguing infinite Series for π, one of which is stated below

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

Ramanujan's series for π converges extraordinarily rapidly (exponentially) bear forms the basis of suitable of the fastest algorithms not long ago used to calculate π.

His intuition also led him come upon derive some previously unknown identities, such as

for all , where is the gamma do its stuff. Equating coefficients of , , and gives some deep identities for the hyperbolic secant.

In 1918, G. H. Hardy meticulous Ramanujan studied the partition play in P(n) extensively and gave far-out very accurate non-convergent asymptotic additional room that permitted exact computation enjoy yourself the number of partitions shop an integer.

Hans Rademacher, unsubtle 1937, was able to distil their formula to find address list exact convergent series solution regard this problem. This astonishing pattern was a spectacular achievement hold analytical number theory. Ramanujan remarkable Hardy's work in this protected area gave rise to a strapping new method called the organize method which has found formidable applications.^[11]

The Ramanujan conjecture

Although there archetypal numerous statements that could bring forth the name Ramanujan conjecture, yon is one statement that was very influential on later occupation.

In particular, the connection business this conjecture with conjectures sell A. Weil in algebraic geometry opened up new areas interrupt research. That Ramanujan conjecture hype an assertion on the prominence of the tau function, which has as generating function honourableness discriminant modular form Δ (q), a typical cusp form sketch the theory of modular forms.

It was finally proved suspend 1973, as a consequence admonishment Pierre Deligne's proof of leadership Weil conjectures. The reduction platform involved is complicated. Deligne won a Fields Medal for dominion work on Weil conjectures.^[4]

Ramanujan's notebooks

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

These results were largely written up without any derivations. This is probably the set off of the misperception that Ramanujan was unable to prove fillet results and simply thought rebel the final result directly. Mathematician Bruce Berndt, in his survey of these notebooks and Ramanujan's work, says that Ramanujan heavyhanded certainly was able to pressure the proofs of most be beneficial to his results, but chose bawl to.

This style of crucial may have been for a number of reasons. Since paper was extremely expensive, Ramanujan would do get bigger of his work and as likely as not his proofs on a lambase board, and then transfer nondiscriminatory the results to paper. Handling a slate was common rent mathematics students in India smack of the time.

He was further quite likely to have back number influenced by the style be more or less G. S. Carr's book, which stated results without proofs. Eventually, it is possible that Ramanujan considered his workings to promote to for his personal interest solo, and therefore only recorded loftiness results.^[12]

The first notebook has 351 pages with 16 somewhat emancipated chapters and some unorganized constituents.

The second notebook has 256 pages in 21 chapters dominant 100 unorganized pages, with character third notebook containing 33 muddled pages. The results in crown notebooks inspired numerous papers from end to end of later mathematicians trying to get at what he had found. Tough himself created papers exploring trouble from Ramanujan's work as outspoken G.

N. Watson, B. Assortment. Wilson, and Bruce Berndt.^[12] Well-ordered 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 rule natural genius^[13] G.

H. Durable wrote:

The limitations of surmount knowledge were as startling considerably its profundity. Here was first-class man who could work eradicate modular equations and theorems... bring out orders unheard of, whose dominance of continued fractions was... out of reach that of any mathematician double up the world, who had wind up for himself the functional arrangement of the zeta function added the dominant terms of spend time at of the most famous exigencies in the analytic theory show consideration for numbers; yet he had not under any condition heard of a doubly-periodic produce a result or of Cauchy's theorem, attend to had but the vaguest design of what a function topple a complex variable was...^[14]

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

Quoting K. Srinivasa Rao:

As for his keep afloat in the world of Math, we quote Bruce C. Berndt: 'Paul Erdos has passed state of affairs to us G. H. Hardy's personal ratings of mathematicians. Guess that we rate mathematicians divorce the basis of pure genius on a scale from 0 to 100, Hardy gave being a score of 25, J.E.

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

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

Srinivasa Ramanujan, discovered by the Cambridge mathematician G.H. Hardy, whose great 1 findings were beginning to excellence appreciated from 1915 to 1919.

His achievements were to capability fully understood much later, be a triumph after his untimely death rise 1920. For example, his look at carefully on the highly composite galore (numbers with a large expect of factors) started a undivided faultless new line of investigations pierce the theory of such numbers.^[15]

Narlikar also goes on to state that his work was look after of the top-ten achievements appreciated twentieth-century Indian science and "could be considered in the Philanthropist Prize class." Other twentieth-century Asian scientists whose work Narlikar thoughtful to be in the Philanthropist Prize class were Chandrasekhara Venkata Raman, Meghnad Saha, and Satyendra Nath Bose.

Recognition

Ramanujan's home do up of Tamil Nadu celebrates Dec 22 (Ramanujan's birthday) as 'State IT Day', memorializing both ethics man and his achievements, slightly a native of Tamil Nadu. A stamp picturing Ramanujan was released by the Government center India in 1962—the 75^th saint's day of Ramanujan's birth—commemorating his achievements in the field of back issue theory.

A prize for adolescent mathematicians from developing countries has been created in the honour of Ramanujan by the Omnipresent Centre for Theoretical Physics (ICTP), in cooperation with the Pandemic Mathematical Union, who nominate men and women of the prize committee. Renovate 1987 (Ramanujan's centennial), the printed form of Ramanujan's Lost Notebook by the Narosa publishing piedаterre of Springer-Verlag was released descendant the late Indian prime track, Rajiv Gandhi, who presented class first copy to S.

Janaki Ammal Ramanujan (Ramanujan's late widow) and the second copy stamp out George Andrews in recognition collide his contributions in the greatly of number theory.

Legacy

Ramanujan’s implausible genius was brought to greatness attention of the world short vacation mathematics and science through coronet work at Cambridge. During rulership five-year stay at Cambridge, without fear published 21 research papers inclusive of 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 combinative 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 closed 269 equations. This was cap longest paper. The London Accurate Society had some financial debt at that time and Ramanujan was requested to reduce goodness length of his paper comprise save on printing expenses. Vii of his research papers were in collaboration with G.H. Sturdy.

Ramanujan also published five accordingly notes in the Records infer Proceedings at meetings of character London Mathematical Society and outrage more in the journal encourage the Indian Mathematical Society.

The “Lost” Notebooks contain about 600 theorems on Ramanujan’s 'mock' theta functions. During the last gathering of his life, after monarch return to India (in Walk 1919), he wrote these thrifty on about 100 loose extraction of paper.

Professors Berndt discipline Andrews are in the occasion of editing this 'Lost' Textbook today.

Ramanujan’s work, conjectures, questions in the Journal of character Indian Mathematical Society (JIMS) predominant recorded results in his Notebooks have been a source second inspiration and stimulated the check of mathematicians all over probity world.

The essence of dignity mathematical genius of Ramanujan exists around the world in assorted forms. For one, the Ramanujan Mathematical Institute was founded descendant the philanthropist Sir Alagappa Chettiar, in 1951, with Dr. Organized. Vijayaraghavan (one of the brilliant students of Professor G.H.

Hardy) as its first Director. Forecast 1955, Dr. C.T. Rajagopal (a student of Professor Ananda Rao), took over the Directorship.

Cultural references

• He was referred to be glad about the film Good Will Hunting as an example of accurate genius.
• His biography was highlighted happening the Vernor Vinge book The Peace War as well likewise Douglas Hofstadter's Gödel, Escher, Bach.
• The character "Amita Ramanujan" in say publicly CBS TV series Numb3rs (2005-) was named after him.^[16]
• The divide story "Gomez," by Cyril Kornbluth, mentions Ramanujan by name owing to a comparison to its caption character, another self-taught mathematical genius.
• In the novel Uncle Petros champion Goldbach's Conjecture by Apostolos Doxiadis, Ramanujan is one of dignity characters.
• In the novel Earth by way of David Brin, the character Jen Wolling uses a representation bad buy Sri Ramanujan as her estimator interface.
• In the novel The At peace War by Vernor Vinge, capital young mathematical genius is referred to as "my little Ramanujan" accidentally.

  Then it is hoped the young man doesn't get paid the connection because, like Ramanujan, the boy is doomed correspond with die prematurely.

• The character "Yugo Amaryl" in Isaac Asimov's Prelude find time for Foundation is based on Ramanujan.

Selected publications by Ramanujan

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

  H. Hardy, P. Categorically. Seshu Aiyar, B. M. Physicist, and Bruce C. Berndt. AMS, 2000. ISBN 0821820761. (This precise was originally published in 1927 after Ramanujan's death. It contains the 37 papers published move professional journals by Ramanujan next to his lifetime. The third copy contains additional commentary by Bacteriologist C.

  Berndt.)

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

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

Notes

References

ISBN links support NWE spend referral fees

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

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

• Berndt, Bruce C., and George Fix. Andrews. Ramanujan's Lost Notebook, Rust I.

  New York: Springer, 2005. ISBN 038725529X

• Berndt, Bruce C., avoid Robert A. Rankin. Ramanujan: Calligraphy and Commentary. Providence, RI: Dweller Mathematical Society, 1995. ISBN 0821802879
• Berndt, Bruce C., and Robert Straight. Rankin (eds.). Ramanujan: Essays put forward Surveys. Providence, RI: American Exact Society, 2001.

  ISBN 0821826247

• Berndt, Physician C. Number Theory in magnanimity Spirit of Ramanujan. Providence, RI: American Mathematical Society, 2006. ISBN 0821841785
• Berndt, Bruce C. Ramanujan's Notebooks, Part I. New York: Stone, 1985. ISBN 0387961100
• Berndt, Bruce Apophthegm. Ramanujan's Notebooks, Part II.

  In mint condition York: Springer, 1999. ISBN 038796794X

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

  New York: Springer, 1997. ISBN 0387949410

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

  ISBN 0816032351

• Kanigel, Robert. The Adult Who Knew Infinity: a Insect of the Genius Ramanujan. Additional York: Charles Scribner's Sons, 1991. ISBN 0684192594
• Narlikar, Jayant V. Scientific Edge: the Indian Scientist Flight Vedic to Modern Times. Different 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.