| 1 | Introductory concepts | Complex numbers, modulus and polar form of a complex number, complex sequences and series, basic topological concepts in the complex plane. |
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| 2 | Complex functions, differentiability and holomorphy | Complex functions, limit and continuity. Exponential function, logarithms and trigonometric functions. Differentiable complex functions, Cauchy-Riemann conditions, holomorphic complex functions. . |
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| 3 | Complex integration | Complex 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 . . |
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| 4 | Power 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. |
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| 5 | Residue theorem and applications | Calculus of Residues and Applications in the calculation of trigonometric and improrer real integrals. |
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| 6 | Conformal functions. | Conformal functions. Mobius transformations and applications in Boundary Value Problems (PDE's). |
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| 7 | Conformal mapping | Conformal mapping. Mobius transformations, Riemann mapping theorem, Schwarz-Christoffel transformation. Applications of conformal mapping. |