The analysis examination (3 hours) covers topics from both real and complex analysis.

It is expected that the student has mastered the basic principles of analysis of real functions of one and several variables. It includes, among other topics, knowledge of the Riemann integral, the differential, the Jacobian matrix, the implicit function theorem, and Stokes' theorem. In addition, the following outlined topics from the theory of functions of a real variable, and the theory of functions of a complex variable should be mastered.

1. Real Analysis. This includes but is not restricted to:
   1. Basic properties of measurable sets and functions.
      1. measurable sets and functions,
      2. Lebesgue measure and Lebesgue integral,
      3. Lebesgue-Stieltjes integral.
   2. Convergence properties of measurable functions and the Lebesgue integral.
      1. algebraic and convergence properties of measurable functions,
      2. convergence theorems for the Lebesgue integral: Fatou's Lemma, monotone convergence theorem, Lebesgue dominated convergence theorem.
   3. Theory of Differentiation. This includes:
      1. functions of bounded variation,
      2. absolutely continuous function and indefinite integrals,
      3. singular functions,
      4. Lebesgue decomposition,
      5. Radon-Nikodym derivative.
   4. Classical Banach and Hilbert Space Theory. This includes:
      1. Banach and Hilbert space
      2. Holder and Minkowski inequalities,
      3. L-p spaces,
      4. representation of bounded linear functions on Hilbert and L-p spaces,
      5. orthonormal families and the Riesz-Fischer theorem,
      6. Fourier series and the Fejer theorem,
      7. linear functions and the Hahn-Banach theorem, Baire's theorem and its consequences: the Banach-Steinhaus, open mapping and closed graph theorem.

Source material for Real Analysis:

1. W. Rudin. Principles of Mathematical Analysis.
2. W. Rudin. Real and Complex Analysis. Chapters 1-6 (from Chapter 2 omit the Riesz representation theorem and from Chapter 5 an abstract approach to the Poisson Integral), Chapter 8 (omit convolutions and distribution functions).
3. H. Royden. Real Analysis. Chapters 2-6 (omit Chapter 4, Section 5), Chapter 10 (omit Sections 5,6,7), Chapter 11 (omit Section 4), Chapter 12 (omit Sections 5,6,7,8,9).

1. Complex Analysis.
   1. Basic algebraic, geometric, and topological properties of complex numbers and complex functions. This includes:
      1. limits and continuity,
      2. exponential, logarithm, and trigonometric functions,
      3. the Riemann sphere,
      4. differentiation and complex line integrals,
      5. the Cauchy-Riemann equations.

   2. Cauchy's integral theorem and its consequences. This includes:
      1. Cauchy's integral theorem,
      2. Cauchy's integral formula for a function and its derivatives and consequences,
      3. the theory of power series and Laurent series,
      4. the theory of zeros and singularities of complex functions, including the residue calculus,
      5. the maximum and identity principles for analytic functions,
      6. the notion of analytic continuation.

   3. Basic theory of harmonic functions. This includes
      1. Laplace equations,
      2. Poisson formula,
      3. maximum principle,
      4. mean value theorem,
      5. the reflection principle.

   4. Basic theory of conformal mapping. This includes
      1. the relation between analytic functions and conformal mapping,
      2. basic examples of conformal mappings, including the theory of linear fractional transformations,
      3. the Riemann mapping theorem.

   Source material for Complex Analysis:

   1. L. Ahlfors. Complex Analysis, McGraw-Hill, Chapters 1-4, Chapter 5, Sections 1 and 5, Chapter 6, Section 1.
   2. E. Hille, Analytic Function Theory, Ginn and Company, Vol. I.
   3. W. Rudin. Real and Complex Analysis, McGraw-Hill, Chapters 10-12, Chapter 14.
   4. R.P. Boas. Invitation to Complex Analysis, Birkhauser, Chapters 1-4.