From Madras to Cambridge: The Journey of Srinivasa Ramanujan

Srinivasa Ramanujan was a brilliant Indian mathematician whose groundbreaking contributions transformed fields such as mathematical analysis, number theory, and infinite series. 

Despite having minimal formal training in mathematics, Ramanujan independently developed sophisticated theories and formulas that astonished even the most prominent mathematicians of his era. His work, known for its depth and elegance, continues to inspire and shape modern mathematical research.

Ramanujan's Master Theorem

One of Ramanujan's significant contributions is the Master Theorem, a powerful tool for evaluating integrals in certain conditions. It connects integrals with the derivatives of a function at zero, providing a bridge between continuous and discrete analysis. The theorem can be stated as:

\[ \int_0^\infty t^{s-1} f(t) \, dt = \Gamma(s) \sum_{n=0}^\infty \frac{(-1)^n}{n!} f^{(n)}(0) \]

Here:

• \(\Gamma(s)\) is the Gamma function, a generalization of the factorial function.
• \(f^{(n)}(0)\) represents the \(n\)-th derivative of \(f\) evaluated at \(t = 0\).

Ramanujan's Approximation to \(\pi\)

Ramanujan discovered many series for approximating \(\pi\), one of which is celebrated for its remarkable speed of convergence:

\[ \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)! (1103 + 26390k)}{(k!)^4 396^{4k}} \]

This series is a cornerstone in numerical calculations of \(\pi\). Each term adds several accurate digits, making it an exceptional method for high-precision computations. Its derivation showcases Ramanujan's profound intuition and creativity in mathematics.

The Taxicab Number

Ramanujan is also famously associated with the taxicab number, stemming from a conversation with the mathematician G.H. Hardy. Hardy mentioned that the number 1729, the license plate of the taxi he had arrived in, seemed uninteresting. Ramanujan immediately countered:

"It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two distinct ways."

\[ 1729 = 1^3 + 12^3 = 9^3 + 10^3 \]

Modular Equations and Dedekind Eta Function

Ramanujan contributed extensively to the theory of modular forms, revealing deep connections between seemingly unrelated mathematical objects. An example of his modular equations involves the Dedekind eta function (\(\eta(\tau)\)), given as:

\[ \frac{\eta(\tau)}{\eta(2\tau)} = \sqrt[4]{2} \sum_{n=0}^\infty \frac{(-1)^n q^{n(3n+1)/2}}{1 + q^{2n+1}} \]

Here:

• \(\eta(\tau)\) is a special function in the study of modular forms.
• \(q = e^{2\pi i \tau}\), where \(\tau\) is a complex number in the upper half-plane.

Legacy

Ramanujan's legacy transcends his individual contributions; his life story serves as a testament to the power of intuition, determination, and genius. From infinite series to modular forms, his work continues to be a beacon of inspiration for mathematicians worldwide.