Bernhard Riemann's contributions to analysis

Bernhard Riemann revolutionized analysis by developing rigorous ways to understand complex functions and curves. He introduced the concept of the Riemann integral, providing a solid foundation for calculus to measure areas under curves precisely. He also invented Riemann surfaces, which allow mathematicians to study multi-valued functions like square roots or logarithms in a unified way. His work connected geometry and analysis, laying groundwork for modern mathematics, including complex analysis, topology, and mathematical physics. Overall, Riemann's ideas transformed how mathematicians analyze continuous phenomena and shape our understanding of space and functions.