Complex analysis - 3AMMA362

Goal(s)

General goals

To acquire a good knowledge of the complex plane properties and to discover the properties and the behaviour of complex variables functions. A particular stress is given to the results and notions frequently used in automatics courses and signal processing courses.

Specific goals

•to know how use complex functions
•to calculate integrals with residue theorem and understand mathematical theories concerning complex fonctions.

Content(s)

1 Complex derivation
1.1 Limits, continuity
1.2 Complex derivation
1.3 Harmonical functions
2 Usual complex functions
2.1 Complex polynomial functions
2.2 N root
2.3 Exponential function, trigonometrical functions,complex logarithm
3.Complex integration
3.1 Line integrals
3.2 Cauchy theorem
3.3 Cauchy integral formula
3.4 Derivatives of a holomorphic function
3.5 Liouville theorem
4. Holomorphic functions analyticity
4.1 Analyticity definition
4.2 Equivalence between analyticity and holomorphic property
4.3 Laurent series
5 Residue theorem and application
5.1 Zeros and singularity
5.2 Residue theorem
5.3 Residue calculation
5.4 Integral calculations with the residue theorem

Mathematics of first and second year, notions of limits, continuity, derivation, partial derivatives, differentiation, integration,line integrals, usual real fonctions, fractions and simple elements decomposition, power series