Real and Complex Analysis

1: Sequences, Series of Functions, and Fourier Analysis
•    Sequences of functions: pointwise and uniform convergence.
•    Series of functions: general functional series and power series.
•    Fourier analysis: Fourier series and their properties.
•    Prerequisites: quick reminders of numeric series.

2: Multivariable Integral Calculus and Differential Geometry
•    Multiple integrals: double and triple Riemann integrals, reduction formulas on cubes, and change of variables (polar, cylindrical, and spherical coordinates).
•    Curves in \(\mathbb{R}^{3}\): length of a curve, rectifiable curves, curve integrals, and their properties.
•    Surfaces in \(\mathbb{R}^{3}\): definition of a surface, regular surfaces, tangent plane, unit normal vector, and surface area.

3: Vector Analysis and Core Field Theorems
•    Vector fields in \(\mathbb{R}^{3}\): definition of vector fields, curl, and divergence.
•    Integrals and fluxes: work of a vector field along a curve, surface integrals, and flux through a surface.
•    Integral theorems: Divergence Theorem and Stokes’ Theorem.
•    Potentials: conservative and irrotational vector fields, simply connected sets, potential existence theorem, and potential construction in \(\mathbb{R}^{2}\) and \(\mathbb{R}^{3}\).

4: Complex Analysis and Integral Transforms
•    Foundations: brief reminders of complex numbers and functions of a complex variable.
•    Holomorphic functions: Cauchy-Riemann conditions, analyticity, and power series.
•    Complex integration: curvilinear integrals in \(\mathbb{C}\), Cauchy-Goursat theorem, and Cauchy formulas.
•    Singularities and residues: Laurent series, zeros of holomorphic functions, isolated singularities, and the Residue Theorem with its applications.
•    Real integrals: evaluation of integrals on the real axis using Jordan's lemma.
•    Integral transforms: definition, properties, and applications of both Fourier and Laplace transforms.