Review of basic notions about the structure of the field of complex numbers. Definition of elementary trascendental function of one complex variable. Holomorphic functions; conditions of Cauchy and Rkiemann, conformal representations. Rules of complex differential calculus. Inverse function, roots, and logarithms. The extended complex plane, and the Riemann sphere. (10h)
Paths in a metric space; regular pahs in the complex plane. Path integrals. Conservative vector fields. Holomorphic functions, as vector fields; the Cauchy theorem. Integrals of the Cauchy type, and Cauchy's integral formula. Harmonic functions. Principle of maximum modulus, and Liouville's theorem.(10 h)
Power series. Analytic functions. Some noteworthy power series. (6h)
Isolated singular points; Laurent expansion; classification of isolated singularities. The Residue Theorem, and its applications. (6h)
The fundamental theorem on Analytic Continuation. Analytic continuation along a path ; monodromy, and polidromy; complete analytic functions, and Riemann surfaces. Integrals of polydromic functions. The Gamma function. (5h)
Linear Ordinary differential equations of 2nd order; existence and uniqueness of local solutions ; analytic continuation of local solutions ; structure of the space of solutions. Singular points: the Euler equation. Fundamental solutions. Regular singularities, and solutions in their neighborhood. The Bessel equation; Bessel functions of 1st and 2nd kind. Equations in the Fuchs class; the equation of Gauss, and the Hypergeometric function. (13h)