S.A.Ramanujan - Srinivasa Aiyangar Ramanujan

S.A.Ramanujan - Died April 26, 1920, Kumbakonam,Born Dec. 22, 1887, Erode, India in Erode, a small village about 400 km southwest of Madras (Chennai). He was one of India's greatest mathematical geniuses. Using one of his identities, mathematics have been able to calculate the value of Π (pi) correct to millions of places of decimal.

Early life

Ramanujan was born on 22 December 1887 in Erode, Tamil Nadu, India, at the place of residence of his maternal grandparents. His father, K. Srinivasa Iyengar worked as a clerk in a sari shop and hailed from the district of Thanjavur. His mother, Komalatammal was a housewife and also a singer at a local temple. They lived in Sarangapani Street in a south-Indian-style home (now a museum) in the town of Kumbakonam. When Ramanujan was a year and a half old, his mother gave birth to a son named Sadagopan. The newborn died less than three months later. In December 1889, Ramanujan had smallpox and fortunately recovered, unlike the thousands in the Thanjavur district who had succumbed to the disease that year. He moved with his mother to her parents' house in Kanchipuram, near Madras. In November 1891, and again in 1894, his mother gave birth, but both children died before their first birthdays.

When he was 15 years old, he obtained a copy of George Shoobridge Carr's Synopsis of elementary results in Pure and Applied Mathematics, 2 vol. (1880–86).This collection of some 6,000 theorems (none of the material was newer than 1860) aroused his genius. Having verified the results in Carr's book, Ramanujan went beyond it, developing his own theorems and ideas. In 1903 he secured a scholarship to the University of Madras but lost it the following year because he neglected all other studies in pursuit of mathematics.

Ramanujan continued his work, without employment and living in the poorest circumstances. After marrying in 1909 he began a search for permanent employment that culminated in an interview with a government official, Ramachandra Rao. Impressed by Ramanujan's mathematical prowess, Rao supported his research for a time, but Ramanujan, unwilling to exist on charity, obtained a clerical post with the Madras Port Trust.

In 1911 Ramanujan published the first of his papers in the Journal of the Indian Mathematical Society. His genius slowly gained recognition, and in 1913 he began a correspondence with the British mathematician Godfrey H. Hardy that led to a special scholarship from the University of Madras and a grant from Trinity College, Cambridge. Overcoming his religious objections, Ramanujan traveled to England in 1914, where Hardy tutored him and collaborated with him in some research.

Ramanujan's knowledge of mathematics (most of which he had worked out for himself) was startling. Although almost completely ignorant of what had been developed, his mastery of continued fractions was unequaled by any living mathematician. He worked out the Riemann series,the elliptic integrals, hyper geometric series, the functional equations of the zeta function, and his own theory of divergent series. On the other hand, the gaps in his knowledge were equally startling. He knew nothing of doubly periodic functions, the classical theory of quadratic forms, or Cauchy's theorem, and had only the most nebulous idea of what constitutes a mathematical proof. Though brilliant, many of his theorems on the theory of prime numbers were completely wrong.

In England Ramanujan made further advances, especially in the partition of numbers. His papers were published in English and European journals, and in 1918 he became the first Indian to be elected to the Royal Society of London.

In 1917 Plagued by health problems all through his life, living in a country far away from home, and obsessively involved with his mathematics, Ramanujan's health worsened in England, perhaps exacerbated by stress, and by the scarcity of vegetarian food during the First World War. He was diagnosed with tuberculosis and a severe vitamin deficiency and was confined to a sanatorium. Ramanujan returned to Kumbakonam, India in 1919 and died soon thereafter at the age of 32,generally unknown to the world at large but recognized by mathematicians as a phenomenal genius, without peer since Leonhard Euler (1707–83)and Karl Jacobi (1804–51).His wife, S. Janaki Ammal, lived in Chennai (formerly Madras) until her death in 1994.

A 1994 analysis of Ramanujan's medical records and symptoms by Dr. D. A. B. Young concluded that it was much more likely he had hepatic amoebiasis, a parasitic infection of the liver. This is supported by the fact that Ramanujan had spent time in Madras, where the disease was widespread. He had two episodes of dysentery before he left India. When not properly treated, dysentery can lie dormant for years and lead to hepatic amoebiasis.It was a difficult disease to diagnose, but once diagnosed would have been readily curable.

Personality

Ramanujan has been described as a person with a somewhat shy and quiet disposition, a dignified man with pleasant manners. He lived a rather spartan life while at Cambridge.

Spiritual life

Ramanujan's first Indian biographers describe him as rigorously orthodox. Ramanujan credited his acumen to his family goddess, Namagiri, and looked to her for inspiration in his work. He often said that an equation for me has no meaning, unless it represents a thought of God.

G. H. Hardy cites Ramanujan as remarking that all religions seemed equally true to him. Hardy further argued that Ramanujan's religiousness had been overstated in the point of belief, not practice by his Indian biographers, and romanticized by Westerners. At the same time, he remarked on Ramanujan's strict observance of vegetarianism.

Mathematical achievements

In mathematics, there is a distinction between having an insight and having a proof. Ramanujan's talent suggested a plethora of formulae that could then be investigated in depth later. It is said that Ramanujan's discoveries are unusually rich and that there is often more in it than what initially meets the eye. As a by-product, new directions of research were opened up. Examples of the most interesting of these formulae include the intriguing infinite series for π, one of which is given below


This result is based on the negative fundamental discriminant d = −4×58 with class number h(d) = 2. Note that 5×7×13×58 = 26390 and is related to the fact that,


Ramanujan's series for π converges extraordinarily rapidly (exponentially) and forms the basis of some of the fastest algorithms currently used to calculate π. Truncating the sum to the first term also gives the approximation for π, which is correct to six decimal places.

One of his remarkable capabilities was the rapid solution for problems. He was sharing a room with P.C.Mahalanobis who had a problem, "Imagine that you are on a street with houses marked 1 through n. There is a house in between (x) such that the sum of the house numbers to left of it equals the sum of the house numbers to its right. If n is between 50 and 500, what are n and x." This is a bivariate problem with multiple solutions. Ramanujan thought about it and gave the answer with a twist: He gave a continued fraction. The unusual part was that it was the solution to the whole class of problems. Mahalanobis was astounded and asked how he did it. "It is simple. The minute I heard the problem, I knew that the answer was a continued fraction. Which continued fraction, I asked myself. Then the answer came to my mind", Ramanujan replied.

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

for all θ, where Γ(z) is the gamma function. Equating coefficients of θ0, θ4, and θ8 gives some deep identities for the hyperbolic secant.

In 1918, G. H. Hardy and Ramanujan studied the partition function P(n) extensively and gave a very accurate non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. Hans Rademacher, in 1937, was able to refine their formula to find an exact convergent series solution to this problem. Ramanujan and Hardy's work in this area gave rise to a powerful new method for finding asymptotic formulae, called the circle method.

One example of his intuition is his discovery of Mock theta functions, in the last year of his life. This was no surprise to some mathematicians as they remarked, "He has his own creativity and the collaboration with Hardy to back it up. So, his finding these is no surprise to the mathematical community." This has gained some interest recently due to proof of the exact formula for the coefficients of any Mock Theta function. Many mathematicians have cited it as the most significant among his discoveries.