Srinivasa Ramanujan: A Glimpse into The

Transcription

1 January 24, 2011 Srinivasa Ramanujan: A Glimpse into The Life of a Self-Taught Mathematical Genius Md. Kamrujjaman An equation means nothing to me unless it expresses a thought of God Ramanujan Department of Mathematics and Statistics University of Calgary

2 Outline Family Backgrounds Academic History and others Contribution in mathematics Ramanujan s Notebooks Conclusion Bibliography References

3 Family Background Born: 22 Dec 1887 in Erode, Tamil Nadu, India Died: 26 April 1920 in Kumbakonam, Tamil Nadu, India Ramanujan belonged to a Brahman family, but his father was poor. His father served as a clerk in a cloth merchant's shop in Kumbakonam. Ramanujan mother was a housewife.

4 Academic History and others Nearly five years old, he entered the primary school. In 1898 at age 10, he entered the Town High School in Kumbakonam. At the age of eleven he was lent books on advanced trigonometry written by S. L. Loney by two lodgers at his home who studied at the Government college. He mastered them by the age of thirteen. He had acquired a first class in his matriculation examination and had also been awarded the Subramanyan scholarship At age of 16 once he obtained a book titled "A Synopsis of Elementary Results in Pure and Applied Mathematics -G. S. Carr (1880). The book generated Ramanujan's interest in mathematics.

5 Academic History and others He was entered to the Government College in Kumbakonam in But he neglected other subjects at the cost of mathematics and failed in college examination. In 1906 Ramanujan went to Madras where he entered Pachaiyappa's College. He passed in mathematics but failed all other subjects. Ramanujan had his first paper published, a 17-page work on Bernoulli numbers that appeared in 1911 in the Journal of the Indian Mathematical Society. In 1912 Ramanujan applied for the post of clerk in the accounts section of the Madras Port Trust and was appointed in the same year.

6 Academic History and others In May 1913, Ramanujan joined the University of Madras as its first research scholar. In the mean time, Ramanujan had approached G. H. Hardy and presented to him a set of 120 theorems and formulas. Hardy and J.E. Littlewood recognized the genious in Ramanujan and made arrangements for him to travel to Cambridge University. At the age of 25 in 1913, he went to Cambridge to continue his work under G. H. Hardy. Over the period of five years, he did most of his work. He is said to have written around 3000 theorems in this time. Most of his work was not as elaborate as a trained mathematicians, because he wasn't.

7 Academic History and others On 1916 Ramanujan graduated from Cambridge with a Bachelor of Science by Research. Ramanujan's dissertation was on Highly composite numbers and seven of his papers published in the Journal of the London Mathematical Society. He often wrote the answer without stating much derivation. Making it difficult for others to see how it was true, but somehow they knew it was coming from such a man. He was elected as a fellow of the Royal Society and also a Fellow of Trinity College, Cambridge and he was the youngest Fellow in the entire history of the Royal Society.

8 Contribution in Mathematics Hardy-Ramanujan number Landau-Ramanujan constant Ramanujan-Soldner constant Ramanujan summation Ramanujan theta function Ramanujan graph Ramanujan s tau function Rogers-Ramanujan identities Ramanujan prime Ramanujan s constant Ramanujan modular functions

9 Hardy-Ramanujan Number Hardy commented that the number 1729 seemed to be uninteresting. Ramanujan is said to have stated on the spot that it was actually a very interesting number mathematically, being the smallest natural number representable as a sum of two cubes in two different ways : = = So far, the following taxicab numbers are known: Ta(1)=2= Ta(2)=1729= = Ta(3)= = = = Up to Ta(8) where Ta(1), Ta(2), Ta(7) and Ta(8) are cube free taxicab numbers i.e T=x 3 +y 3 with x,y relatively prime.

10 Rogers-Ramanujan identities The Rogers Ramanujan identities are given by where Application: The Rogers-Ramanujan identities appeared in Baxter's solution of the hard hexagon model in statistical mechanics. It s a 2-dimensional lattice model of a gas, where particles are allowed to be on the vertices of a triangular lattice but no two particles may be adjacent.

11 Rogers-Ramanujan identities Lattice models are popular in theoretical and computational physics for many reasons. Some models are exactly solvable and some other with perturbation theory. Triangular tilling. The vertices form a hexagonal lattice with horizontal rows, with triangles pointing up and down.

12 Ramanujan modular functions If, then and are modular functions of T, where G and H are Rogers-Ramanujan identities. One of a modular function is the Dedikind eta function where the relation between the Dedikind eta function and Ramanujan tau function are If one substitutes q = exp(2πiz), the function with z>0, known as discriminant modular form.

13 Ramanujan modular functions Modular functions are used in the mathematical analysis of Riemann surfaces. Riemann surface theory is relevant to describing the behavior of strings as they move through space-time. When a string moves in space-time by splitting and recombining, a large number of mathematical identities must be satisfied. These are the identities of Ramanujan s modular function. The "Ramanujan function has 24 "modes" that correspond to the physical vibrations of a bosonic string. When the Ramanujan function is generalized, 24 is replaced by 8 for fermion strings.

14 Ramanujan s summation Ramanujan summation is a technique for assigning a sum to infinite divergent series. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties which make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined. Ramanujan summation essentially is a property of the partial sums, rather than a property of the entire sum, as it doesn't exist.

15 Ramanujan s summation If we take the Euler-Maclaurin summation formula together with the correction rule using Bernoulli numbers, we see that: Ramanujan wrote it for the case p going to infinity: Where C is a constant specific to the series.

16 Ramanujan s summation Comparing both formulae and assuming that R tends to 0 as x tends to infinity, we see that, in a general case, for functions f(x) with no divergence at x = 0: where Ramanujan assumed a = 0. By taking we normally recover the usual summation for convergent series. For functions f(x) with no divergence at x = 1, we obtain: C(0) was then proposed to use as the sum of the divergent sequence.

17 Ramanujan s summation In particular, the sum of where the notation indicates Ramanujan summation. This formula originally appeared in one of Ramanujan's notebooks, without any notation to indicate that it was a Ramanujan summation. For even powers we have: For odd powers we have a relation with the Bernoulli numbers: Those values are consistent with the Riemann zeta function.

18 Ramanujan s summation Ramanujan found several rapidly converging infinite series ofπas

19 Ramanujan s Notebook While still in India, Ramanujan recorded the bulk of his results in four notebooks of loose leaf paper. These results were mostly written up without any derivations. Since paper was very expensive, Ramanujan would do most of his work and perhaps his proofs on slate, and then transfer just the results to paper. The first notebook has 351 pages with 16 somewhat organized chapters and some unorganized material. The second notebook has 256 pages in 21 chapters and 100 unorganized pages, with the third notebook containing 33 unorganized pages. A fourth notebook with 87 unorganized pages, the so-called lost notebook.

20 Lost Notebook Lost notebook is the manuscript in which Ramanujan, from Cambridge University, recorded the mathematical discoveries of the last year of his life. It was rediscovered by George Andrews in 1976, in a box of effects of G. N. Watson stored at the Wren Library at Trinity College, Cambridge. The "notebook contains with more than 600 of Ramanujan's formulas. The majority of the formulas are about q-series, mock theta functions, modular equations, and the remaining formulas are mainly about integrals, Dirichlet series, congruences, and asymptotics. It has been claimed that Ramanujan's modular functions provide a rationale for string theory being expressed in ten dimensions.

21 Image of Notebook and Bust

22 Conclusion Ramanujan was a self-taught mathematician.the story is one of the most tragic, romantic and haunting in all of mathematics. It is the story of how a genius, by his sheer brilliance and supreme faith in his own ability, became a true scientific immortal. Further, as Hardy notes, the real tragedy of Ramanujan was not his early death at the age of 32, but that in his most formative years, he did not receive proper training, and so a significant part of his work was rediscovery....

23 Bibliography G H Hardy, Ramanujan (Cambridge, 1940). B C Berndt and R A Rankin, Ramanujan : Letters and commentary (Providence, Rhode Island, 1995). S R Ranganathan, Ramanujan : the man and the mathematician (London, 1967). S Ram, Srinivasa Ramanujan (New Delhi, 1979). P V Seshu Aiyar, The late Mr S Ramanujan, B.A., F.R.S., J. Indian Math. Soc. 12 (1920), B Berndt, Srinivasa Ramanujan, The American Scholar 58 (1989), E H Neville, Srinivasa Ramanujan, Nature 149 (1942),

24 References Number theory in the spirit of Ramanujan by Bruce C. Berndt nivasa-ramanujan The man who knew infinity: a life of the genius Ramanujan by Robert Kanigel

25 Courtesy Thanks to All