Analytic continuation is a mathematical technique used to extend the domain of a given analytic function beyond its original region. In quantum field theory (QFT), it plays a crucial role in connecting different physical theories and resolving singularities. The process ensures that if a function is analytic on an open subset of the complex plane, it can be uniquely defined outside that set while maintaining its analytic properties. This concept highlights the interplay between mathematics and physics, emphasizing the importance of rigorous mathematical foundations in theoretical physics. Understanding analytic continuation is essential for advanced studies in QFT and related fields.