Università Roma Tre

Complex numbers: algebraic and topological properties. Geometric representation of complex numbers: polar coordinates and the complex exponential.
Complex functions with complex variables: continuity and properties, differentiability and first properties. Holomorphic functions: properties and examples of holomorphic and non-holomorphic functions. Cauchy-Riemann equations. The real and imaginary parts of holomorphic functions are harmonic conjugated.
Equations of Cauchy-Riemann: proof. Examples. Sequences and complex series. Properties. Power series with complex values. Abel's theorem and Hadamard's formula.
Proof of Abel's Theorem. Taylor's formula for series of complex powers. The exponential and the trigonometric functions as analytical functions. Basic properties.
Periodicity of the complex exponential function. The complex logarithm: first considerations. The ring of formal powers series with complex coefficients: basic properties. Analytical functions: definition and first properties.
Series of converging powers are analytical within the convergence region. Composition of analytical functions. Theorem of the inverse function.
Inverse by composition of a formal series and its convergence. Complex powers and properties. The binomial series and properties. Consequences of the inverse theorem: the canonical form of an analytic function.
Local properties of analytical functions: open function theorem, invertibility criterion, principle of the maximum local module. The fundamental theorem of algebra. Parameterized curves. A holomorphic function with zero derivative is constant.
The locus of the zeros of a non-constant analytical function is discrete. Analytical continuation of functions defined on open connected sets. Principle of the maximum global module. Integrals in paths: definition and first properties. Examples.
A continuous function in a connected open admits a primitive if and only if its integral along a closed curve is zero. Integration of uniformly converging series of functions. Examples. Local primitive of a holomorphic function.
Local primitive of a holomorphic function. The Goursat theorem. Integral of a holomorphic function along a continuous path.
The homotopical form of the Cauchy Theorem. Global primitive of a holomorphic function in a simply connected domain. Applications to the study of the logarithm.
The integral formula of Cauchy. Cauchy formula for development in series and applications: a holomorphic and analytical function; the theorem of Liouville and the fundamental theorem of algebra.
Integral formula for derivatives. The number of windings of a curve with respect to a point. Curves homologous to 0. The global formula of Cauchy.
Proof of the global Cauchy formula. Examples.
The first homology group of an open set with values ​​in integers. The Cauchy formula for homological invariance. Examples.
Applications of the Cauchy theorem: uniform limit on holomorphic function compacts is holomorphic. Examples. Laurent series.
Series expansion of a holomorphic function in a circular crown in the Laurent series. Isolated singularities and the field of meromorphic functions. Examples. Statement of the classification theorem of isolated singularities and residual theorem: local and global versions.
Proof of the classification theorem of isolated singularities and proof of the residues theorem. The logarithmic derivative and the principle of the argument. Calculation of residues.
Classification of the connected open of C. The Riemann map theorem and the uniformization theorem (without proof). The Riemann sphere as a compactification of the complex plane. The group of linear transformations of the projective line and the linear transformations produced by them. The group of automorphisms of the complex plane.
The lemma of Schwarz and the group of automorphisms of the unitary disc. Elements of global analytical functions and function. The logarithm as a global analytical function.
The n-th rooty as a global analytical function. The bundle of germs of analytical functions and its properties. The Riemann surface associated with a global analytical function.
Examples and properties of Riemann surface. The Riemann surface associated with an algebraic function and properties. Summary and considerations on the course program.

L. V. Ahlfors: Complex Analysis, McGraw-Hill.
S. Lang: Complex analysis, GTM 103.
E. Freitag, R. Busam: Complex Analysis, Springer.