The content: basics on limits and continuous functions of one clomplex variable; curves in the complex plane and integration; derivative; Cauchy integral formula and consequences: analyticity of holomorphic functions, Theorems of Morera, Weierstrass (on sequence of holomorphic functions which converge uniformly on compact sets), Montel, Maximum modulus and Liouville. Schwarz reflection principle. Isolated singularities and residus. Argument principle, theorems of Hurwitz and Rouche; conformal transformations and Riemann Theorem. Harmonic functions, theorem of the mean, Gauss theorem, Poisson formula in the disk. Application to the Diriclet problem for the Laplace equation. Elements of Laplace transform. The text is written in Italian. |