Description
1Introductory conceptsComplex numbers, modulus and polar form of a complex number, complex sequences and series, basic topological concepts in the complex plane.
2Complex functions, differentiability and holomorphyComplex functions, limit and continuity. Exponential function, logarithms and trigonometric functions. Differentiable complex functions, Cauchy-Riemann conditions, holomorphic complex functions. .
3Complex integrationComplex contour integral. Cauchy-Goursat theorem, Deformation Principle, Cauchy's Integral formulas, Liouville's theorem, Maximum Modulus Principle, harmonic functions, uniqueness of the solution to the Poisson problem . .
4Power and Laurent series, singularities.Power series and convergence radius. Taylor's theorem and Taylor's expansions of basic complex numbers. Laurent series and isolated singularities: removable singularities, poles and essential singularities.
5Residue theorem and applicationsCalculus of Residues and Applications in the calculation of trigonometric and improrer real integrals.
6Conformal functions.Conformal functions. Mobius transformations and applications in Boundary Value Problems (PDE's).
7Conformal mappingConformal mapping. Mobius transformations, Riemann mapping theorem, Schwarz-Christoffel transformation. Applications of conformal mapping.