Università Roma Tre

Reminders on complex numbers: algebraic and topological properties, the compactification to a point of the field of complex numbers and its identification with the sphere via stereographic projection, the polar representation of complex numbers.
Holomorphic functions: examples (polynomials, rational functions, exponential), non-examples (the complex conjugation), Cauchy-Riemann equations.
The algebra of the formal powers series: order of a power series, inverse, composition, derivative,
The quotient field of the domain of the formal series: the field of Laurent's series, residuals, inverse.
Convergence: punctual, absolute, uniform and uniform on the compact. The geometric series and its convergence properties.
Abel's Theorem for the convergence of power series: convergence radius, absolute and uniform convergence on compacts inside the convergence disk, divergence on the outside, holomorphicity of the limit function, convergence of the derived series.
Series of converging powers and holomorphic functions associated to them. Convergence of the derivative of a convergent series, of the sum and product of convergent series, of the convergent series composition, of the multiplicative inverse and of the inverse by composition.
Analytical functions: local series development of an analytical function; the analytic functions are infinitely derivable and all their derivatives are analytic; sum, product, inverse and composition of analytical functions are analytical; the analytic functions have a local analytic primitive, unique up to a constant, a convergent series defines an analytic function in its convergence disk, the inverse function theorem.
Normal form of an analytic function: each analytical function is locally, up tp translation and change of coordinates, the elevation to a power or is locally constant.
The open function theorem. Criterion of analytical isomorphism. Principle of the local and global maximum.
The zeros of an analytical function are discrete. Locally constant functions are constant in a connected open subset. Proof of the fundamental theorem of algebra (using the principle of the global maximum module).
The integral of a complex continuous function along a piecewise C^1-curve.
Criterion for the existence of a primitive: a continuous function admits a primitive if and only if its integral is zero along any closed curve. Example: the complex exponential as an analytic function.
The integral of uniformly convergent sequences and series.
A holomorphic function in a disk admits a primitive.
The integral of a holomorphic function along a continuous curve (not necessarily C^1).
The homotopy between continuous curves. The homotopic invariance theorem of the integral. Corollary: A holomorphic function on a simply connected open admits a primitive.
The integral (or local) formula of Cauchy (without proof). Cauchy formula for the series expansion of a holomorphic function (without proof). Corollaries: holomorphic functions are analytic, a power series is not holomorphic in at least one point of the boundary of the convergence disk, a whole function with an infinite polynomial expansion is a polynomial, a bounded entire function is constant (theorem of Liouville), integral formula of Cauchy for derivatives of a holomorphic function. Application: proof of the fundamental theorem of algebra using the Liouville theorem.
Proof of the integral (or local) formula of Cauchy. Proof of the Cauchy formula for the series expansion of a holomorphic fuction.
Winding number of a closed curve around a point: definition and examples.
Properties of the winding number of a closed curve: analytical definition, constancy in the connected components of the curve complement.
Homologous zero curves in an open subset. Relationship between homotopy and homology. The global formula of Cauchy: Dixon proof.
Chains and cycles. Integration along a chain. Winding number of a cycle. Zero homologous cycles. The first group of homology (with integral coefficients) of an open and its relation with the fundamental group. Examples.
The global Cauchy formula and Cauchy's Theorem on homological invariance: their equivalence.
The proof of Cauchy's Theorem on homological invariance (taken from the book by Alfhors).
Sequences of holomorphic functions uniformly converging on compacts: holomorphicity of the limit function; the derivative and the integral on a chain can be exchanged with the limit.
Laurent series (infinite) converging and holomorphic functions on a circular annulus.
Isolated singularities: removable singularities, poles, essential singularities. Examples. Rational functions. The theorems (by Riemann and Casorati-Weierstrass) of characterization of isolated singularities.
The residual theorem: local and global version.
Meromorphic functions and their properties. The logarithmic derivative of a meromorphic function and its properties. The principle of the argument. Corollary: number of zeros and poles within a simple closed curve. Application: proof of the fundamental Theorem of Algebra using the principle of the argument.
Roche's theorem. Application: proof of the fundamental Theorem of Algebra using Roche's theorem.
An approach to the problem of classification of domains through the map theorem and the uniformization theorem (without proof). Examples: the disk and the upper half-plane are biolomorphic (Cayley transform), the universal coating of C ^ * is the complex plane and the covering map is the exponential.
The complex projective line as a compactification of the complex plane. The complex projective line is homeomorphic to the sphere through stereographic projection. The linear projective group PGL2 acts on the complex projective line by means of linear (or Moebius) linear transformations.
The group of automorphisms of the complex plane.
Schwarz's Lemma. The group of automorphisms of the unitary disk.
The classification of the subgroups of the automorphisms of the complex plane that act in a free and properly discontinuous manner.
Coating quotients of the complex plane.
Elliptic (or doubly periodic) functions with respect to a lattice. The only holomorphic elliptic functions are constants. Properties of zeros and poles of elliptic functions inside a fundamental domain.
The Weiertrass P function and its derivative P ': definition and convergence properties.
Properties of the P function and its derivative P ': ellipticity, parity / disparity, poles and zeros.
The polynomial relationship between P and P '. All elliptic functions are expressed as rational functions in P and P '. Corollary: the field of elliptic functions.

S. Lang: Complex Analysis. Fourth edition. Graduate Texts in Mathematics, 103. Springer-Verlag, New York, 1999.
L. V. Ahlfors: Complex Analysis. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York, 1978.
R. Shakarchi: Problems and solutions for complex analysis. Springer-Verlag, New York, 1999.