ANALYTICAL STUDY OF RAMANUJAN’S INFINITE SERIES AND SPECIAL FUNCTIONS IN MODERN MATHEMATICS

Abstract

The mathematical discoveries of Srinivasa Ramanujan continue to occupy a central position in modern mathematical research due to their originality, depth, and far-reaching implications. Among his most remarkable contributions are his studies on infinite series, special functions, continued fractions, and asymptotic expansions. This paper presents an analytical study of Ramanujan’s infinite series and special functions using contemporary mathematical approaches. The research focuses on selected identities from Ramanujan’s notebooks and examines their mathematical structure, convergence properties, and theoretical significance.

The study reconstructs important results using methods from Complex Analysis, Real Analysis, and Number Theory. Special attention is given to Ramanujan’s treatment of divergent series, gamma functions, zeta functions, and modular relations. The paper further explores how these results anticipated modern developments in asymptotic analysis and mathematical physics.

The findings reveal that Ramanujan’s methods, though highly intuitive, align closely with advanced analytical frameworks developed later in the twentieth century. His infinite series transformations and special function identities continue to influence modern research in theoretical mathematics, quantum physics, and computational analysis. The study concludes that Ramanujan’s work represents a unique synthesis of intuition and analytical depth, making his contributions enduringly relevant in contemporary mathematical sciences.