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Introduction:

Srinivasa Ramanujan, a mathematical prodigy from India, made profound contributions to the fields of mathematical analysis and number theory during the early 20th century. His work, marked by extraordinary depth and originality, has significantly influenced the historical development of mathematics.

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Contributions to Mathematical Analysis

• Infinite Series and Pi: Ramanujan’s formulas for the calculation of Pi (π) were groundbreaking. His series for π, which converges extraordinarily rapidly, revolutionized the way π is calculated. 
• Partition Function: He developed formulas to calculate the partition function p(n), which represents the number of distinct ways of expressing a positive integer n as a sum of positive integers. His asymptotic formula for p(n) is an exemplary blend of analysis and number theory.

Contributions to Number Theory

• Ramanujan Primes: He conceptualized ‘Ramanujan Primes’, a new class of prime numbers, which have properties significant in understanding prime number distributions. These primes have applications in cryptography and computer science.
• Ramanujan Conjecture: His conjecture on the tau-function, linked to the coefficients of the power series of the modular form, was a significant hypothesis that influenced the development of algebraic number theory and was later proved by Pierre Deligne as a part of his proof of the Weil conjectures.
• Highly Composite Numbers: Ramanujan introduced the concept of Highly Composite Numbers, which are integers with more divisors than any smaller number. This concept has applications in fields like cryptography and coding theory.

Significance in Historical Development

Ramanujan’s work symbolizes a remarkable blend of intuition and rigor, lacking formal training in higher mathematics. His notebooks, filled with results, have been a treasure trove for mathematicians, leading to further advancements and discoveries. 

His contributions:

• Inspired Further Research: Many of his findings were seeds for new theories, such as the Hardy-Ramanujan asymptotic formula, which led to significant development in analytic number theory.
• Cross-Cultural Scientific Collaboration: His collaboration with G.H. Hardy serves as a classic example of cross-cultural scientific cooperation, setting a precedent for future international collaborations in the academic world.
• Influence on Modern Mathematics: Concepts like “mock theta functions” have found relevance in modern-day physics, particularly in string theory, showcasing the enduring impact of his work.

Conclusion:

Srinivasa Ramanujan’s contributions to mathematical analysis and number theory are monumental. His intuitive approach, coupled with rigorous mathematical insights, not only advanced the field of mathematics but also paved the way for future generations of mathematicians. His legacy transcends time, continuing to inspire and challenge the mathematical community. Ramanujan’s life and work epitomize the universal language of mathematics and its boundless potential, irrespective of one’s formal training and origin.

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