| 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. |
