Freitag, Eberhard., Busam, Rolf. Complex Analysis. Germany: Springer Berlin Heidelberg, 2009
The Identity theorem is a fundamental concept in complex analysis that states that if two analytic functions defined on an open set have the same values on a sequence of points that have a limit point in the set, then the two functions are equal on the entire open set. In other words, the values of an analytic function on a certain set uniquely determine the function itself.
This theorem is widely used in complex analysis to prove the uniqueness of analytic functions, which are functions that can be represented as a convergent power series in a complex variable. It also plays a crucial role in the study of Riemann's zeta function and Riemann's Last Theorem, as it allows us to show that the zeta function has only one analytic continuation. This property is crucial in the proof of Riemann's Last Theorem and other important theorems in number theory.
Overall, the Identity theorem is a powerful tool in complex analysis, allowing us to establish the uniqueness of analytic functions and providing a foundation for many important results in mathematics.
Despite identity theorem simplicity, this theorem is incredibly powerful and finds frequent applications in our research.