Ramanujan And His Contributions To Mathematics.

Born in a small village in the southwest of Madras, India ,Srinivasa Ramanujan was one of India's greatest mathematical geniuses. Though he left the world at a very young age on 26th April, 1920, Still , in a short life span of just 32 years, he succeeded in making some important contributions to the advancement of mathematics. Here are some of the most significant contributions made by the genius to the world of mathematics.

Srinivasa Ramanujan

Ramanujan Summation

Ramanujan did interesting mathematics in the field of infinite summation. He invented Ramanujan summation, which is a technique for assigning a value to divergent infinite series. It is essentially a property of the partial sums, rather than a property of the entire sum. This method for summation of numbers, points to the fact that ‘S’= -1/12, where
S = 1+2+3+4+5+6+7+……..
Though this result is shocking, but, string theory, quantum field theory and some complex analytics make extensive use of this to derive equations.

Related: Life lessons that we can learn from the life of Ramanujan

Highly Composite Numbers

A highly composite number is basically a positive integer which has more divisors than any smaller positive integer. Ramanujan coined this term in 1915. There are an infinite number of highly composite numbers, the first few being 1, 2, 4, 6, 12, 24, 36, 48, 60…and so on. The corresponding numbers of divisors are 1, 2, 3, 4, 6, 8, 9, 10, 12… and so on. In 1915, Ramanujan listed 102 highly composite numbers up to 6746328388800. Later in 1983 and 1988, Robin & Nicholas modified this list.

Ramanujan's Master Theorem

Ramanujan's master theorem is related to Analytic functions and Mellin transforms. A Mellin transform is an integral transform that can be regarded as the multiplicative version of the two-sided Laplace transform.And, Analytic function is a function that is locally given by a convergent power series. Ramanujan's master theorem is basically a technique that provides an analytic expression for the Mellin transform of an analytic function. According to this,
The result is stated as follows:

If a complex-valued function f(x) is expanded as -
f(x) = ∑ φ(k)/k! (-x)^k …for k ranging from 0 to infinity,
Then, the Mellin transform of f(x) is given by -

Integral[x^(s-1)f(x)]dx = Γ(s)φ(-s)

[integration limit is 0 to infinity] , Γ(s) represents gamma function. Ramanujan used this widely to calculate definite integrals and infinite series. This theorem is also of great importance in quantum mechanics.

Hardy Ramanujan Number

When G.H. Hardy came to see Ramanujan in a taxi numbered 1729, G.H Hardy stated "Ramanujan that 1729 seemed to be a very dull number, and hoped it doesnt turn out to be an unfavorable omen. But, thanks to his love for numbers, Ramanujan found something special about this number as well and said "it is a very interesting number, 1729 is the smallest number which can be written in the form of sum of cubes of two numbers in two ways,i.e 1729=1³+12³=9³+10³".Since than the number 1729 is called Hardy-Ramanujan’s number.

Related: The unique bond of Hardy And Ramanujan

Apart from this, He also made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, hypergeometric series, infinite series and many other fields. Ramanujan also provided solutions to some of mathematical problems,earlier considered unsolvable.

Ramanujan(centre), G.H Hardy (extreme right), with other scientists outside the Senate House, Cambridge
Image Courtesy - Wikipedia

When asked about Ramanujan to G. H Hardy, the famous English mathematician stated-

"The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems…to orders unheard of, whose mastery of continued fraction was… beyond that of any mathematician in the world, who had found for himself the functional equation of zeta function and the dominant terms of many of the most famous problems in analytical theory of numbers; and yet he had never heard of a doubly periodic function or of Cauchy’s theorem, and had indeed but the vaguest idea of what a function of complex variable was…”

Ramanujan never received a proper formal training in pure mathematics, still he made substantial contributions to various fields of mathematics. This makes him one of the most determined, intellectual and inspiring mathematicians to look up to!

Related: Leonhard Euler - The man who changed the course of mathematics.

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