General information, questionnaire, and start with advanced calculus:
Elementary inequalities, such as Cauchy's inequality and AM-GM inequality, Series: convergence tests for positive series such as comparison test, limit comparison test, ratio test, root test
Some standard series, Cauchy condensation test, recognizing a Riemann sum, telescoping series, (generalized) Dirichlet test, power series and Cauchy-Hadamard Theorem, approximation to the identity, methods to interchange limits and integrals such as Dominated Convergence Theorem.
Multivariable calculus: Derivatives, partial derivatives, mixed partial derivatives, optimization and Lagrange multipliers and applications to prove inequalities, Inverse and Implicit Function Theorems, Green's Theorem, Stokes Theorem, Divergence Theorem, Fourier Series
Change of basis, matrix representation of linear maps, Jordan decomposition, rank-nullity theorem.
Matrix Decompositions: Polar, SVD, LU, Choleski, QR, Schur.
Powers of matrices, positive definite matrices, characterization of commutators, Gram matrix, projections.
Complex numbers, complex functions, complex differentiability, holomorphic functions, conformal maps, Cauchy-Riemann equations, (formal) power series, Cauchy-Hadamard, open mapping and inverse function theorem, maximum principles, path integration, Cauchy-Goursat, primitives of holomorphic functions.
Cauchy Formula, Laurent Series, singularities, Casorati-Weierstrass, analytic automorphisms of complex numbers, Residue formula, Rouche's theorem.
Calculation of integrals by Residue Theorem, evaluation of series by Residue Theorem, Conformal mappings, Fractional linear transformations.Exercise Sheet Complex Variables 2
More on conformal maps, Weierstrass representation theorem, Hadamard's theorem.Exercise Sheet Complex Variables 3
Analytic continuation, Schwarz reflection, Blaschke products. Work in class on exercises from exam Fall 2007.
Final remarks, some interesting exercises, finish exam Fall 2007.