# Written exam workshop Spring 2012

## Resources

• First, it is very important to get a copy of the past written exams from Tamar's office.
• There is a Written Exam Wiki with solutions to almost all of the past written exam problems: good if you want to check your answers or find out how to do a problem on which you're stuck.
• I would advise you to find at least one good reference on each subject, from which you can review the theory. What is a good reference is a matter of personal taste, pick a book that appeals to you and that contains enough information for preparing for the exam. You can find a few suggestions below.
• There is various information that you can find on websites of previous written exam workshops:
• Andrew Suk has written up solutions to exam problems

## Tentative Schedule

### February 7 - Introduction and 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 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.