Srinivasa Ramanujan Iyengar
Total Page：16
File Type：pdf, Size：1020Kb
Srinivasa Ramanujan Iyengar Srinivasa Ramanujan Iyengar (Srinivāsa Rāmānujan Iyengār)() (December 22, 1887 – April 26, 1920) was an Indian mathematician widely regarded as one of the greatest mathematical minds in recent history. With almost no formal training in mathematics, he made profound contributions in the areas of analysis and number theory. A child prodigy, Ramanujan was largely self-taught and compiled nearly 3,884 theorems during his short lifetime. Although a small number of these theorems were actually false, most of his statements have now been proven to be correct. His deep intuition and uncanny algebraic manipulative ability enabled him to state highly original and unconventional results that have inspired a vast amount of research; however, some of his discoveries have been slow to enter the mathematical mainstream. Recently his formulae have begun to be applied in the field of crystallography and physics. The Ramanujan Journal was launched specifically to publish work "in areas of mathematics influenced by Ramanujan". With the encouragement of friends, he wrote to mathematicians in Cambridge seeking validation of his work. Twice he wrote with no response; on the third try, he found the English Mathematician G. H. Hardy. Ramanujan's arrival at Cambridge (March 1914) was the beginning of a very successful five-year collaboration with Hardy. One remarkable result of the Hardy-Ramanujan collaboration was a formula for the number p(n) exp(π 2n /3) of partitions of a number n: pn()∼ asn→∞.A partition of a positive integer n is just an expression for n as a sum of 43n positive integers, regardless of order. Thus p(4) = 5 because 4 can be written as 1+1+1+1, 1+1+2, 2+2, 1+3, or 4. The problem of finding p(n) was studied by Euler, who found a formula for the generating function of p(n) (that is, for the infinite series whose nth term is p(n)xn). While this allows one to calculate p(n) recursively, it does not lead to an explicit formula. Hardy and Ramanujan came up with such a formula (though they only proved it works asymptotically; Rademacher proved it gives the exact value of p(n)). “Srinivasa Ramanujan”. <http://www.usna.edu/Users/math/meh/ramanujan.html> “Srinivasa Ramanujan”. <http://en.wikipedia.org/wiki/Srinivasa_Ramanujan> .