Written exam workshop
Spring 2012



Advanced Calculus

Linear Algebra

Complex Analysis

Tentative Schedule

February 7 - Introduction and Advanced Calculus

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

February 14 - Advanced Calculus

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.

February 21 - Advanced Calculus

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

February 28 - Linear Algebra

Change of basis, matrix representation of linear maps, Jordan decomposition, rank-nullity theorem.

Exercise Sheet Linear Algebra 1

March 6 - Linear Algebra

Matrix Decompositions: Polar, SVD, LU, Choleski, QR, Schur.

Exercise Sheet Linear Algebra 2

March 13 - No workshop: Spring break

March 20 - Linear Algebra

Powers of matrices, positive definite matrices, characterization of commutators, Gram matrix, projections.

Exercise Sheet Linear Algebra 3

March 27 - Complex Variables

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.

April 3 - Complex Variables

Cauchy Formula, Laurent Series, singularities, Casorati-Weierstrass, analytic automorphisms of complex numbers, Residue formula, Rouche's theorem.

Exercise Sheet Complex Variables 1

April 10 - Complex Variables

Calculation of integrals by Residue Theorem, evaluation of series by Residue Theorem, Conformal mappings, Fractional linear transformations.

Exercise Sheet Complex Variables 2

April 17

More on conformal maps, Weierstrass representation theorem, Hadamard's theorem.

Exercise Sheet Complex Variables 3

April 24 - Exercises from exam Fall 2007

Analytic continuation, Schwarz reflection, Blaschke products. Work in class on exercises from exam Fall 2007.

May 1 - Exercises from exam Fall 2007

Final remarks, some interesting exercises, finish exam Fall 2007.