Srinivasa Ramanujan

It's a tale from the annals of mathematics. G. H. Hardy, an English mathematician, got a bizarre letter from an anonymous clerk in Madras, India, in 1913. About 120 theorems on infinite series, improper integrals, continuing fractions, and number theory were presented in the ten-page letter. Every renowned mathematician receives letters from cranks, and Hardy no certainly placed this letter in that category at first look. But something about the formulas drew his attention, prompting him to present it to his partner J. E. Littlewood. After a few hours, they came to the conclusion that the results "must be real since no one would have had the creativity to develop them if they weren't true."

 

Srinivasa Ramanujan (1887-1920) was born into a family of mathematicians. Ramanujan, who was born in South India, was a bright student who won academic awards in high school. However, after obtaining a book titled A Synopsis of Elementary Results in Pure and Applied Mathematics at the age of 16, his life took a significant turn. The book was only a collection of thousands of mathematical findings, most of which were presented with little or no evidence. It was not intended to be a mathematical classic; rather, it was created to assist English mathematics students who were preparing for the infamously tough Tripos test, which required extensive memorizing. But it sparked a frenzy of mathematical activity in Ramanujan, who labored through the book's results and beyond. Unfortunately, Ramanujan's complete concentration in mathematics was terrible for his scholastic career: he frequently failed his college examinations by disregarding all of his other courses. Ramanujan's situation was precarious as a college dropout from an impoverished household. He survived on the kindness of strangers, filling notebooks with mathematical discoveries and looking for sponsors to fund his studies. Finally, when the Indian mathematician Ramachandra Rao supplied him with a tiny grant and subsequently a clerkship at the Madras Port Trust, he had some success. During this time, Ramanujan published his first paper, a 17-page study on Bernoulli numbers, in the Journal of the Indian Mathematical Society in 1911. No one knew if Ramanujan was a true genius or a knucklehead. With the help of friends, he wrote to Cambridge mathematicians seeking confirmation of his work. He wrote twice and received no answer; on the third attempt, he discovered Hardy.

 

Hardy responded warmly to Ramanujan's letter, and Ramanujan's position rose almost immediately as a result of Hardy's endorsement. Ramanujan was appointed a research scholar at the University of Madras, earning twice as much as a clerk and only having to give quarterly reports on his work. Hardy, on the other hand, was adamant that Ramanujan be brought to England. Ramanujan's mother first objected—high-caste Indians eschewed travel to foreign lands—but eventually relented, purportedly after a vision. Ramanujan boarded a ship bound for England in March 1914. Ramanujan's presence at Cambridge marked the start of a five-year partnership with Hardy that was a huge success. Hardy was a strong proponent of rigor in analysis, but Ramanujan's conclusions were "arrived at by a process of mixed reasoning, intuition, and induction, of which he was completely unable to provide any cohesive explanation," as Hardy put it. Hardy did his best not to discourage Ramanujan by filling up the gaps in his education. "I have never seen his equal, and can only compare him to Euler or Jacobi," he said of Ramanujan's remarkable formal acumen in handling infinite series, continuing fractions, and the like.

 

A formula for the number p(n) of partitions of a number n was one of the most notable outcomes of the Hardy-Ramanujan cooperation. A positive integer n partition is just an equation for n as a sum of positive integers, in any order. Because 4 may be represented as 1+1+1+1, 1+1+2, 2+2, 1+3, or 4, p(4) = 5. Euler investigated the issue of calculating p(n) and discovered a formula for the generating function of p(n) (that is, for the infinite series whose nth component is p(n)xn). While this allows one to iteratively compute p(n), it does not result in an explicit formula. Hardy and Ramanujan devised such a formula (albeit only asymptotically; Rademacher demonstrated that it yields the precise value of p(n)).

 

Ramanujan's time in England was fruitful in terms of mathematics, and he received the acclaim he deserved. In 1916, he received a Bachelor of Science degree "by research" from Cambridge, and in 1918, he was elected a Fellow of the Royal Society (the first Indian to do so). His health, however, was harmed by the strange environment and society. Ramanujan had always lived in a tropical environment and had his mother (later his wife) cook for him; now he had to cook everything himself in order to follow his caste's rigorous dietary requirements. Wartime shortages exacerbated the situation. His physicians feared for his life when he was admitted to the hospital in 1917. His health had recovered by late 1918, and he went to India in 1919. However, his condition deteriorated once again, and he died the next year.

 

Apart from his published works, Ramanujan left behind many notebooks that have been extensively studied. G. N. Watson, an English mathematician, authored a series of articles about them. Bruce C. Berndt, an American mathematician, has just published a multi-volume analysis of the notebooks. The Ramanujan Journal was founded in 1997 with the goal of publishing work "in areas of mathematics affected by Ramanujan."

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