Programme

Complex numbers and geometric representation.
Complex exponential.
Complex functions of complex variables: continuity and differentiability.
Holomorphic functions. Cauchy-Riemann equations.
Sequences and complex series.
Power series with complex values.
Abel's theorem and Hadamard's formula.
Taylor's formula for series of complex powers.
The exponential and the trigonometric functions as analytical functions.
The complex logarithm.
The ring of formal powers series with complex coefficients.
Analytic functions.
Inverse function theorem.
Complex powers.
The binomial series and properties.
Canonical form of an analytic function.
Open function theorem.
Invertibility criterion.
Principle of the maximum local module.
The fundamental theorem of algebra.
A holomorphic function with zero derivative is constant.
The locus of the zeros of a non-constant analytical function is discrete.
Principle of the maximum global module.
Integrals along paths.
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.
Local primitive of a holomorphic function.
The Goursat theorem.
The homotopical form of Cauchy Theorem.
Global primitive of a holomorphic function.
Application to the logarithm.
Cauchy's integral formula
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 winding number of winding of a curve with respect to a point.
The global formula of Cauchy.
The first homology group of an open set.
The Cauchy formula for homological invariance.
Applications of the Cauchy theorem: uniform limit on holomorphic function compacts is holomorphic.
Laurent series.
Series expansion of a holomorphic function in a circular crown in t Laurent series.
Isolated singularities and the field of meromorphic functions.
Classification theorem of isolated singularities and the residue theorem.
The logarithmic derivative and the principle of the argument. Calculation of residues.
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.
he group of automorphisms of the complex plane.
Schwarz lemma and the group of automorphisms of the unitary disc.
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.

Core Documentation

L. V. Ahlfors: Complex Analysis, McGraw-Hill.
S. Lang: Complex analysis, GTM 103.

Attendance

In class

Type of evaluation

Written exam with exercises and oral exam on the theory.

Programme

Complex numbers and geometric representation.
Complex exponential.
Complex functions of complex variables: continuity and differentiability.
Holomorphic functions. Cauchy-Riemann equations.
Sequences and complex series.
Power series with complex values.
Abel's theorem and Hadamard's formula.
Taylor's formula for series of complex powers.
The exponential and the trigonometric functions as analytical functions.
The complex logarithm.
The ring of formal powers series with complex coefficients.
Analytic functions.
Inverse function theorem.
Complex powers.
The binomial series and properties.
Canonical form of an analytic function.
Open function theorem.
Invertibility criterion.
Principle of the maximum local module.
The fundamental theorem of algebra.
A holomorphic function with zero derivative is constant.
The locus of the zeros of a non-constant analytical function is discrete.
Principle of the maximum global module.
Integrals along paths.
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.
Local primitive of a holomorphic function.
The Goursat theorem.
The homotopical form of Cauchy Theorem.
Global primitive of a holomorphic function.
Application to the logarithm.
Cauchy's integral formula
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 winding number of winding of a curve with respect to a point.
The global formula of Cauchy.
The first homology group of an open set.
The Cauchy formula for homological invariance.
Applications of the Cauchy theorem: uniform limit on holomorphic function compacts is holomorphic.
Laurent series.
Series expansion of a holomorphic function in a circular crown in t Laurent series.
Isolated singularities and the field of meromorphic functions.
Classification theorem of isolated singularities and the residue theorem.
The logarithmic derivative and the principle of the argument. Calculation of residues.
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.
he group of automorphisms of the complex plane.
Schwarz lemma and the group of automorphisms of the unitary disc.
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.

Core Documentation

L. V. Ahlfors: Complex Analysis, McGraw-Hill.
S. Lang: Complex analysis, GTM 103.

Attendance

In class

Type of evaluation

Written exam with exercises and oral exam on the theory.

Programme

Complex numbers and geometric representation.
Complex exponential.
Complex functions of complex variables: continuity and differentiability.
Holomorphic functions. Cauchy-Riemann equations.
Sequences and complex series.
Power series with complex values.
Abel's theorem and Hadamard's formula.
Taylor's formula for series of complex powers.
The exponential and the trigonometric functions as analytical functions.
The complex logarithm.
The ring of formal powers series with complex coefficients.
Analytic functions.
Inverse function theorem.
Complex powers.
The binomial series and properties.
Canonical form of an analytic function.
Open function theorem.
Invertibility criterion.
Principle of the maximum local module.
The fundamental theorem of algebra.
A holomorphic function with zero derivative is constant.
The locus of the zeros of a non-constant analytical function is discrete.
Principle of the maximum global module.
Integrals along paths.
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.
Local primitive of a holomorphic function.
The Goursat theorem.
The homotopical form of Cauchy Theorem.
Global primitive of a holomorphic function.
Application to the logarithm.
Cauchy's integral formula
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 winding number of winding of a curve with respect to a point.
The global formula of Cauchy.
The first homology group of an open set.
The Cauchy formula for homological invariance.
Applications of the Cauchy theorem: uniform limit on holomorphic function compacts is holomorphic.
Laurent series.
Series expansion of a holomorphic function in a circular crown in t Laurent series.
Isolated singularities and the field of meromorphic functions.
Classification theorem of isolated singularities and the residue theorem.
The logarithmic derivative and the principle of the argument. Calculation of residues.
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.
he group of automorphisms of the complex plane.
Schwarz lemma and the group of automorphisms of the unitary disc.
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.

Core Documentation

L. V. Ahlfors: Complex Analysis, McGraw-Hill.
S. Lang: Complex analysis, GTM 103.

Attendance

In class

Type of evaluation

Written exam with exercises and oral exam on the theory.