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Complex Analysis and The Fellowship of the Identity Theorem

Today I wanted to talk about a simple theorem from Complex Analysis that really only needs a bit of Calculus 2 knowhow to understand, but has big implications. Complex Analysis is one of the core subjects that come up time and again in Analysis, even in the study of Operator Theory, where it is used to show that the spectrum of an operator is nonempty. One particularly useful tool in Function Theoretic Operator Theory is the Identity Theorem, which is a surprisingly simple theorem to prove. I personally use this theorem (and its multivariate generalization) all throughout my research in operator theory, and we talked about it briefly in my videos talking about generalizations of the SINDy algorithm. A week ago or so, I posted about a "Strangely Deep Problem About Sequences," which was pulled from Paul Halmos' "A Hilbert Space Problem Book." There we used the identity theorem to show that the Szego kernel (the kernel of the Hardy space) was dense in the Hardy space itself. So, I thought it'd be instructive for my students (and any of you that are interested) to make a video going over the identity theorem. This is a quick video and should be accessible to anyone with a bit of a calculus background. Music: (Forthcoming)