What is schwarz's theorem and proved it?

Schwarz's theorem, also known as Schwarz's reflection principle, states that if a function is analytic in a domain and continuous on its closure, then if it takes real values on a boundary segment, it can be extended to an analytic function across that segment by reflecting it. This means that the function can be mirrored across the boundary, maintaining its properties in the extended domain. The proof involves demonstrating that the reflected function remains analytic and satisfies the necessary conditions in the extended region. This theorem is fundamental in complex analysis and has applications in various fields, including physics and engineering.