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Mathematics In he was hospitalized, his doctors fearing for his life. By late his health had improved; he returned to India in But his health failed again, and he died the next year.
O Ramanujan was appointed to the post of clerk and began his duties on 1stMarch He sent theorems on imply divisibility properties of the partition function.
Ramanujan and his associates had shown that every large integer could be written as the sum of at most four Example: Partition of whole numbers: Partition of whole numbers is another similar problem that captured ramanujan attention.
Subsequently ramanujan developed a formula for the Custom essay for o india of any number, which can be made to yield the required result by a series of successive approximation. Ramanujan studied the highly composite numbers also which are recognized as the opposite of prime numbers.
He studies their structure, distribution and special forms. Ramanujam was shown how to solve cubic equations in and he went on to find his own method to solve the quadratic.
The following year, not knowing that the quintic could not be solved by radicals, he tried and of course failed to solve the quintic. By Ramanujam had began to undertake deep research.
He worked hypo geometric series, and investigated relations between integrals and series. He was to discover later that he had been studying elliptic functions. Journal of the Indian mathematical society: Ramanujan continued to develop his mathematical ideas and began to pose problems and solve problems in the journal of the Indian mathematical society.
He developed relations between elliptic modular equations in He published a brilliant research paper on Bernoulli numbers in in the journal of the Indian mathematical society and gained recognition for his work. Despite his lack of a university education, he was becoming well known in the madras area as a mathematical genius.
He began to study the Bernoulli numbers, although this was entirely his own independent discovery. Ramanujan worked out the Riemann series, the elliptic integrals hyper geometric series and functions equations of the zeta functions on the other hand he had only a vague idea of what constitutes a mathematical proof.
Despite many brilliant results, some of his theorems on prime numbers were completely wrong.
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Perhaps has most famous work was on the number p n for small numbers n, and ramaujan used this numerical data to conjecture some remarkable properties some of which he proved using elliptic functions. In a joint paper with hardly, ramanujan gave an asymptotic formulas for p n.
It had the remarkable property that it appeared to give the correct value of p nand this was later proved by Rademacher. Ramanujan discovered a number of remarkable identities that imply divisibility properties of the partition function.
He also produced quite a number of results in definite integrals in the form of general formulate. Besides his published work, ramanujan left behind several notebooks filled with theorems that mathematicians have continued to study. The English Mathematician G. N Watson, from topublished 14 papers under the general title theorems stated by Ramanujan and in all he published nearly 30 papers which were inspired by ramanjan work.C ontrast Study: An imaging procedure in which a contrast medium is introduced to enhance the image of a particular body region or structure..
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