Riemann surfaces and function theory

Riemann surfaces are mathematical structures that allow complex functions—functions involving square roots, logarithms, etc.—to be understood without ambiguity. Since some functions behave differently depending on the path taken around certain points, Riemann surfaces provide a way to "unfold" these functions onto multi-layered surfaces, making their behavior smooth and consistent. Function theory on these surfaces studies the properties of such complex functions, exploring how they behave, transform, and connect across different regions, helping mathematicians understand intricate patterns and structures in complex analysis.