Works of Ramanujan

By- Shreet MishraClass-XI (2nd shift)Roll no.- 47

Introduction

Srinivasa Ramanujan (22 December 1887 –26 April 1920) was an Indian mathematicianand autodidact who, with almost no formaltraining in pure mathematics, madeextraordinary contributions to mathematicalanalysis, number theory, infinite series,and continued fractions. Ramanujan initiallydeveloped his own mathematical research inisolation, which was quickly recognized byIndian mathematicians.

Introduction

When his skills became apparent to the widermathematical community, centered in Europeat the time, he began a famous partnershipwith the English mathematician G. H. Hardy.He rediscovered previously known theoremsin addition to producing new work.Ramanujan was said to be a natural genius, inthe same league as mathematicians suchas Euler and Gauss.

Introduction

Ramanujan independently compiled nearly 3900

results (mostly identities and equations). Nearlyall his claims have now been proven correct,although a small number of these results wereactually false and some were already known. Hestated results that were both original and highlyunconventional, such as the Ramanujanprime and the Ramanujan theta function, andthese have inspired a vast amount of furtherresearch.

Early Life & His Introduction To Mathematics

On 1 October 1892, Ramanujan was enrolled at the local school. In March 1894, he was moved to a Telugu medium school. At the Kangayan Primary School, Ramanujan performed well. Just before the age of 10, in November 1897, he passed his primary examinations in English, Tamil, geography and arithmetic. By age 11, he had exhausted the mathematical knowledge of two college students who were lodgers at his home. He was later lent a book on advanced trigonometry written by S. L. Loney. He completely mastered this book by the age of 13 and discovered sophisticated theorems on his own.

Early Life & His Introduction To Mathematics

He completed mathematical exams in half the allotted time, and showed a familiarity with geometry and infinite series.By 14, he was receiving merit certificates and academic awards which continued throughout his school career and also assisted the school in the logistics of assigning its 1200 students (each with their own needs) to its 35-odd teachers.

Early Life & His Introduction To Mathematics

Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic. The following year, not knowing that the quintic could not be solved by radicals, he tried (and of course failed) to solve the quintic.

Calculation Euler–MascheroniConstant

In 1903 when he was 16, Ramanujan obtained from afriend a library-loaned copy of a book by G. S. Carr. Thebook was titled A Synopsis of Elementary Results inPure and Applied Mathematics and was a collection of5000 theorems. Ramanujan reportedly studied thecontents of the book in detail. The book is generallyacknowledged as a key element in awakening thegenius of Ramanujan. The next year, he hadindependently developed and investigatedthe Bernoulli numbers and had calculated the Euler–Mascheroni constant up to 15 decimal places.

Ramanujan & IMC• Ramanujan met deputy collector V. Ramaswamy

Aiyer, who had recently founded the IndianMathematical Society (IMC). Ramanujan, wishingfor a job at the revenue department whereRamaswamy Aiyer worked, showed him hismathematics notebooks. As Ramaswamy Aiyerlater recalled:

• I was struck by the extraordinary mathematicalresults contained in it [the notebooks]. I had nomind to smother his genius by an appointment inthe lowest rungs of the revenue department

Ramanujan & Bernoulli numbers

One property he discovered was that the denominators (sequence A027642 in OEIS) of the fractions of Bernoulli numbers were always divisible by six. He also devised a method of calculating Bn based on previous Bernoulli numbers. One of these methods went as follows:It will be observed that if n is even but not equal to zero,(i) Bn is a fraction and the numerator of in its lowest terms is a prime number,(ii) the denominator of Bn contains each of the factors 2 and 3 once and only once,(iii) is an integer and consequently is an odd integer.

Ramanujan & His Journal

One of the first problems he posed in the journal was:

He waited for a solution to be offered in threeissues, over six months, but failed to receiveany. At the end, Ramanujan supplied thesolution to the problem himself.

Ramanujan at London Mathematical Society

Ramanujan was awarded aBachelor of Science degree byresearch (this degree was laterrenamed PhD) in March 1916 forhis work on highly compositenumbers, the first part of whichwas published as a paper inthe Proceedings of the LondonMathematical Society. The paperwas over 50 pages with differentproperties of such numbersproven.

Ramanujan’s mathematic philosophy

In mathematics, there is a distinction between having an

insight and having a proof. Ramanujan's talent suggesteda plethora of formulae that could then be investigated indepth later. It is said by G. H. Hardy that Ramanujan'sdiscoveries are unusually rich and that there is oftenmore to them than initially meets the eye. As a by-product, new directions of research were opened up.Examples of the most interesting of these formulaeinclude the intriguing infinite series for π, one of which isgiven below

Valuable Works Of Ramanujan

Landau–Ramanujan constantMock theta functionsRamanujan conjectureRamanujan primeRamanujan–Soldner constantRamanujan theta functionRamanujan's sumRogers–Ramanujan identitiesRamanujan's master theorem

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