Smirnov's Theorem is a result in complex analysis that deals with the convergence of sequences of analytic functions. It states that if a sequence of analytic functions converges uniformly on compact subsets of a domain, then the limit function is also analytic in that domain. This theorem is significant because it provides a way to interchange limits and integrals, which is crucial in many areas of mathematical analysis. The theorem is named after the Russian mathematician Nikolai Smirnov, who contributed to the field of functional analysis. It is often used in the study of holomorphic functions and plays a key role in understanding the behavior of complex functions in various mathematical contexts.