• Math 645: Analysis I
• Math 745: Analysis II
• Math 656: Complex Variables I  

Material Covered 

The real analysis portion of the exam focuses on material covered in the two one-semester courses `Analysis I and Analysis II' (Math 645 and Math 745) and in chapters three through eleven of the second edition of `Mathematical Analysis' by T.M. Apostol. For example, students should be familiar with the theory of the Riemann integral (but not necessarily the Riemann-Stieltjes integral), the Fundamental Theorem of Calculus, and the Lesbesgue criterion for Riemann integrability. Another good reference for the Riemann integral is `Calculus' by M. Spivak.

The complex analysis portion of the exam focuses on material covered in the one-semester course Complex Variables I (Math 656). This corresponds approximately to chapters 1 through 5 of `Complex Variables' by Ablowitz and Fokas or chapters 1 through 4 of `Functions of a Complex Variable' by Carrier, Krook and Pearson.  

Students should note that the exam emphasizes an ability to apply the material described in the outline.

Detailed Outline, Real Analysis

 Basic Topics

1. Elementary logic and set theory, including countable and uncountable sets
2. Axiomatic formulation of the real number system, including the axiom of completeness
3. Elementary properties of the integers, rational, and complex numbers
4. Complex arithmetic and algebra, including the Fundamental Theorem of Algebra
5. Elementary complex functions
6. Open sets, closed sets, and limit points in metric spaces
7. The Bolzano-Weierstrass and Heine-Borel theorems
8. Compact sets and their properties
9. Complete spaces and their properties
10. Convergence of a sequence in a metric space
11. The limit of a function (single variable)
12. Continuity and uniform continuity of a function
13. The derivatives of a real-valued function and their properties, including the Mean-Value Theorem and Taylor's Theorem

Advanced Calculus

1. Convergence of real-valued series and their products
2. Differentiation of real-valued and vector-valued functions on R
3. Implicit functions
4. Extrema

Theory of the Integral

1. The Riemann integral, including the Fundamental Theorem of Calculus and Lebesgue's criterion for Riemann integrability
2. The Lebesgue integral
3. Convergence theorems and their applications for the Lebesgue integral
4. Measure and measurable functions

Basic Functional Analysis

1. Uniform convergence
2. The space of square integrable functions,
3. Fixed points of contraction mappings

Basic Harmonic Analysis

1. Convergence of Fourier series
2. Properties of Fourier integrals
3. The Poisson summation formula

Detailed Outline, Complex Analysis

1. Complex numbers, function of a complex variable, limit and continuity
2. Analytic functions and the Cauchy-Riemann equations
3. Multi-valued functions, branch points, basic consideration of Riemann surfaces, and branch cuts
4. The integral of a complex function. Cauchy's theorem and Cauchy's integral formula
5. Liouville's theorem, Morera's theorem. The maximum (minimum) modulus theorem and its implication for a harmonic function
6. Complex series. Power, Taylor, and Laurent series. Radius of convergence
7. Singularities of complex functions. Zeros, poles, and essential singularities
8. Calculus of residues and examples of complex contour integrals
9. Principle of the argument, Rouche's theorem
10. Conformal mapping, including applications to problems of two-dimensional potential theory. Mapping theorems and examples of conformal mappings (bilinear and Schwartz-Christoffel mappings)
11. Mittag-Leffler theorem
12. Simple applications to transform theory (Fourier and Laplace transforms)

Copies of past qualifying exams are available here.