Complex Analysis I

Graduate complex analysis builds the rigid, almost magical theory of functions that are complex-differentiable on an open set, a hypothesis so strong that one derivative forces infinitely many, local behavior controls global behavior, and integration reduces to counting poles. You'll work through problem sets that move from Cauchy's theorem and residue calculus into the deeper structural results: the argument principle, conformal equivalence via the Riemann mapping theorem, and harmonic function theory through Poisson integrals. It's the analytic backbone for later courses in Riemann surfaces, several complex variables, and analytic number theory, and the place where most students first see how rigidity can be more powerful than generality.